Evgueni Tevelev

Projectively Dual Varieties

Preface

During several centuries various reincarnations of projective duality have inspired research in algebraic and differential geometry, classical mechanics, invariant theory, combinatorics, etc. On the other hand, projective duality is simply the systematic way of recovering the projective variety from the set of its tangent hyperplanes. In this survey we have tried to collect together different aspects of projective duality and points of view on it. We hope, that the exposition is quite informal and requires only a standard knowledge of algebraic geometry and algebraic (or Lie) groups theory. Some chapters are, however, more difficult and use the modern intersection theory and homology algebra. But even in these cases we have tried to give simple examples and avoid technical difficulties.

An interesting feature of projective duality is given by the observation that most important examples carry the natural action of the Lie group. This is especially true for projective varieties that have extremal properties from the point of view of projective geometry. We have tried to stress this phenomenon in this survey and to discuss many variants of it. However, one aspect is completely omitted – we are not discussing the dual varieties of toric varieties and the corresponding theory of A-discriminants. This theory is presented in the beautiful book [GKZ2] and we feel no need to reproduce it.

Parts of this survey were written during my visits to the Erwin Shroedinger Institute in Vienna and Mathematic Institute in Basel. I would like to thank my hosts for the warm hospitality. I have discussed the contents of this book with many people, including E. Vinberg, V. Popov, A. Kuznetsov, S. Keel, H. Kraft, D. Timashev, D. Saltman, P. Katsylo, and learned a lot from them. I am especially grateful to F. Zak for providing a lot of information on projective duality and other aspects of projective geometry.

Edinburgh, November 2001 Evgueni Tevelev

II Preface

Table of Contents

1. Dual Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.1 Definitions and First Properties . . . . . . . . . . . . . . . . . . . . . . . . . . 1 1.2 Reflexivity Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.A The Conormal Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4 1.2.B Applications of the Reflexivity Theorem . . . . . . . . . . . . . 6

1.3 Dual Plane Curves . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8 1.3.A Parametric Representation of the Dual Plane Curve . . 8 1.3.B The Legendre Transformation and Caustics . . . . . . . . . . 9 1.3.C Correspondence of Branches. Plücker Formulas. . . . . . . 10

1.4 Projections and Linear Normality . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.A Projections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12 1.4.B Linear Normality . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

1.5 Dual Varieties of Smooth Divisors . . . . . . . . . . . . . . . . . . . . . . . . 17

2. Dual Varieties of Algebraic Group Orbits . . . . . . . . . . . . . . . . . 21

2.1 Polarized Flag Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.A Definitions and Notations . . . . . . . . . . . . . . . . . . . . . . . . . 21 2.1.B Basic Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

2.2 The Pyasetskii Pairing . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30 2.2.A Actions With Finitely Many Orbits . . . . . . . . . . . . . . . . . 30 2.2.B The Multisegment Duality. . . . . . . . . . . . . . . . . . . . . . . . . 32

2.3 Parabolic Subgroups With Abelian Unipotent Radical . . . . . . . 34

3. The Cayley Method for Studying Discriminants . . . . . . . . . . 41 3.1 Jet Bundles and Koszul Complexes . . . . . . . . . . . . . . . . . . . . . . . 41 3.2 Cayley Determinants of Exact Complexes . . . . . . . . . . . . . . . . . 43 3.3 Discriminant Complexes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46

4. Resultants and Schemes of Zeros . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.1 Ample Vector Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 51 4.2 Resultants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52 4.3 Zeros of Generic Global Sections . . . . . . . . . . . . . . . . . . . . . . . . . 54 4.4 Moore–Penrose Inverse and Applications . . . . . . . . . . . . . . . . . . 61 5. Fulton–Hansen Theorem and Applications . . . . . . . . . . . . . . . . 71

5.1 Fulton–Hansen Connectedness Theorem . . . . . . . . . . . . . . . . . . . 71 5.2 Secant and Tangential Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 74 5.3 Zak Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

6. Dual Varieties and Projective Differential Geometry . . . . . . 81 6.1 The Katz Dimension Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81 6.2 Product Theorem and Applications . . . . . . . . . . . . . . . . . . . . . . . 84

6.2.A Product Theorem . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84 6.2.B Hyperdeterminants . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87 6.2.C Associated Hypersurfaces . . . . . . . . . . . . . . . . . . . . . . . . . . 88

6.3 Ein Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 89 6.4 The Projective Second Fundamental Form . . . . . . . . . . . . . . . . . 96

6.4.A Gauss Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96 6.4.B Moving Frames . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98 6.4.C Fundamental Forms of Projective Homogeneous Spaces 101

7. The Degree of a Dual Variety . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.1 Katz–Kleiman–Holme Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . 105 7.2 Degrees of Discriminants of Polarized Flag Varieties . . . . . . . . 108

7.2.A The Degree of the Dual Variety to G/B . . . . . . . . . . . . . 108 7.2.B Degrees of Hyperdeterminants . . . . . . . . . . . . . . . . . . . . . 110 7.2.C One Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 111

7.3 Degree of the Discriminant as a Function in L. . . . . . . . . . . . . . 116 7.3.A General Positivity Theorem . . . . . . . . . . . . . . . . . . . . . . . . 117 7.3.B Applications to Polarized Flag Varieties . . . . . . . . . . . . . 118

7.4 Gelfand–Kapranov Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 119

8. Milnor Classes and Multiplicities of Discriminants . . . . . . . . 121 8.1 Class Formula . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 121 8.2 Milnor Classes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122 8.3 Applications to Multiplicities of Discriminants . . . . . . . . . . . . . 124 8.4 Multiplicities of the Dual Variety of a Surface . . . . . . . . . . . . . . 126 8.5 Further Applications to Dual Varieties . . . . . . . . . . . . . . . . . . . . 128

9. Mori Theory and Dual Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 133 9.1 Some Results From Mori Theory . . . . . . . . . . . . . . . . . . . . . . . . . 133 9.2 Mori Theory and Dual Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . 138

9.2.A The Nef Value and the Defect . . . . . . . . . . . . . . . . . . . . . . 138 9.2.B The Defect of Fibers of the Nef Value Morphism . . . . . 140 9.2.C Varieties With Small Dual Varieties. . . . . . . . . . . . . . . . . 142

9.3 Polarized Flag Varieties With Positive Defect . . . . . . . . . . . . . . 144 9.3.A Nef Value of Polarized Flag Varieties . . . . . . . . . . . . . . . . 144 9.3.B Classification . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147 10. Some Applications of the Duality . . . . . . . . . . . . . . . . . . . . . . . . . 151

10.1 Discriminants and Automorphisms. . . . . . . . . . . . . . . . . . . . . . . . 151 10.1.A Matsumura–Monsky Theorem . . . . . . . . . . . . . . . . . . . . . . 151 10.1.B Quasiderivations of Commutative Algebras . . . . . . . . . . 153

10.2 Discriminants of Anticommutative Algebras . . . . . . . . . . . . . . . . 157 10.3 Self–dual Varieties . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 168

10.3.A Self–dual Polarized Flag Varieties . . . . . . . . . . . . . . . . . . 168 10.3.B Around Hartshorne Conjecture . . . . . . . . . . . . . . . . . . . . . 170 10.3.C Self–dual Nilpotent Orbits . . . . . . . . . . . . . . . . . . . . . . . . . 172

10.4 Linear Systems of Quadrics of Constant Rank . . . . . . . . . . . . . . 172

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 178

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 187

1. Dual Varieties

Preliminaries

The projective duality gives a remarkably simple method to recover any projective variety from the set of its tangent hyperplanes. In this chapter we recall this classical notion, give the proof of the Reflexivity Theorem and its consequences, provide a number of examples and motivations, and fix the notation to be used throughout the book. The exposition is fairly standard and classical. In the proof of the Reflexivity Theorem we follow [GKZ2] and deduce this result from the classical theorem of symplectic geometry saying that any conical Lagrangian subvariety of the cotangent bundle is equal to some conormal variety.

1.1 Definitions and First Properties

For any finite-dimensional complex vector space V we denote by ℙ(V) its projectivization, that is, the set of 1-dimensional subspaces. For example, if V = ℂ^{n+1} is a standard complex vector space then ℙ^{n} = ℙ(ℂ^{n+1}) is a standard complex projective space. A point of ℙ^{n} is defined by (n + 1) homogeneous coordinates (x_{0} : …: x_{1}), x_{i} ∈ ℂ, which are not all equal to 0 and are considered up to a scalar multiple.

If U ⊂ V is a non-trivial linear subspace then ℙ(U) is a subset of ℙ(V), subsets of this form are called projective subspaces. Projective subspaces of dimension 1, 2, or of codimension 1 are called lines, planes, and hyperplanes.

For any vector space V we denote by V^{∗} the dual vector space, the vector space of linear forms on V. Points of the dual projective space ℙ(V)^{∗} = ℙ(V^{∗}) correspond to hyperplanes in ℙ(V). Conversely, to any point p of ℙ(V), we can associate a hyperplane in ℙ(V)^{∗}, namely the set of all hyperplanes in ℙ(V) passing through p. Therefore, ℙ(V)^{∗∗} is naturally identified with ℙ(V). Of course, this reflects nothing else but a usual canonical isomorphism V^{∗∗} = V.

To any vector subspace U ⊂ V we associate its annihilator Ann(U) ⊂ V^{∗}. Namely, Ann(U) = {f ∈ V^{∗} | f(U) = 0}. We have Ann(Ann(U)) = U. This corresponds to the projective duality between projective subspaces in ℙ(V) and ℙ(V)^{∗}: for any projective subspace L ⊂ ℙ(V) we denote by L^{∗} ⊂ ℙ(V)^{∗} its dual projective subspace, parametrizing all hyperplanes that contain L.

Remarkably, the projective duality between projective subspaces in ℙ^{n} and ℙ^{n}^{∗} can be extended to the involutive correspondence between irreducible algebraic subvarieties in ℙ^{n} and ℙ^{n}^{∗}.

First, suppose that X ⊂ ℙ^{n} is a smooth irreducible algebraic subvariety. For any x ∈ X, we denote by T̂_{x}X ⊂ ℙ^{n} an embedded projective tangent space. More precisely, if X ∈ ℙ(V) is any projective variety then we define the cone Cone(X) ⊂ V over it as a conical variety formed by all lines l such that ℙ(l) ∈ X. If x ∈ X is a smooth point then any non-zero point v of the corresponding line is a smooth point of Cone(X) and T̂_{x}(X) is defined as ℙ(T_{v} Cone(X)), where T_{v} Cone(X) is a tangent space of Cone(X) at v considered as a linear subspace of V (it does not depend on a choice of v). For any hyperplane H ⊂ ℙ^{n}, we say that H is tangent to X at x if H contains T̂_{x}X. We define the dual variety X^{∗} ⊂ ℙ^{n}^{∗} as the set of all hyperplanes tangent to X.

In other words, a hyperplane H belongs to X^{∗} if and only if the intersection X ∩ H (regarded as a scheme) is singular (is not a smooth algebraic variety). In most parts of this book we shall be interested only in dual varieties of smooth varieties, so this description of X^{∗} will be sufficient. However, with this definition we can not expect the duality X^{∗∗} = X because X^{∗} can be singular (and in most interesting cases it is actually singular). So we should define X^{∗} for a singular X as well. There are in fact two possibilities: first, we can imitate the previous definition and consider embedded tangent spaces at all points, not necessarily smooth. But it turns out that the dual variety defined in this fashion does not have good properties, for example it can be reducible. The better way is to pick only the ‘main’ component of the dual variety.

Definition 1.1 Let X ⊂ ℙ^{n} be an irreducible projective variety. A hyperplane H ⊂ ℙ^{n} is called tangent to X if it contains an embedded tangent space T̂_{x}X at some smooth point x ∈ X. The closure of the set of all tangent hyperplanes is called the dual variety X^{∗} ⊂ ℙ^{n}^{∗}.

We shall discuss the Reflexivity Theorem X^{∗∗} = X and its consequences in the next section. First we shall establish some simple properties of dual varieties and give further definitions.

Definition 1.2 Let X ⊂ ℙ^{n} be an irreducible projective variety with the smooth locus X_{sm}. Consider the set I_{X}^{0} ⊂ ℙ^{n} × ℙ^{n}^{∗} of pairs (x, H) such that x ∈ X_{sm} and H is the hyperplane tangent to X at x. The Zariski closure I_{X} of I_{X}^{0} is called the conormal variety of X.

The projection pr_{1} : I_{X}^{0} → X_{sm} makes I_{X}^{0} into a bundle over X_{sm} whose fibers are projective subspaces of dimension n − dim X − 1. Therefore, I_{X}^{0} and I_{X} are irreducible varieties of dimension n − 1. By definition, X^{∗} is the image of the projection pr_{2} : I_{X} → ℙ^{n}^{∗}. Therefore, we have the following proposition:

Proposition 1.3 X^{∗} is an irreducible variety.

Moreover, since dim I_{X} = n − 1, we can expect that in ‘typical’ cases X^{∗} is a hypersurface. Having this in mind, we give the following definition:

Definition 1.4 The number codim_{ℙ^{n}}^{∗} X^{∗} − 1 is called the defect of X, denoted by def X.

Typically, def X = 0. In this case X^{∗} is defined by an irreducible homogeneous polynomial Δ_{X}.

Definition 1.5 Δ_{X} is called the discriminant of X.

If def X > 0 then, for convenience, we set Δ_{X} = 1. Clearly, Δ_{X} is defined only up to a scalar multiple. Roughly speaking, the study of dual varieties and discriminants includes the following 3 steps:

– To define some nice natural class of projective varieties X.

– To find all exceptional cases when def X > 0.

– In the remaining cases, to say something about Δ_{X} or X^{∗}.

In the last step the minimal program is to determine the degree of Δ_{X}, the maximal program is to determine Δ_{X} as a polynomial, or at least to describe its monomials. Another interesting problem is to describe singularities of X^{∗}.

Example 1.6 The most familiar example of a discriminant Δ_{X} is, of course, the discriminant of a binary form. In order to show that it actually coincides with some Δ_{X} we need first to give an equivalent definition of Δ_{X}. Suppose that x_{1}, …, x_{k} are some local coordinates on Cone(X) ⊂ V. Any f ∈ V^{∗}, a linear form on V, being restricted to Cone(X) becomes an algebraic function in x_{1}, …, x_{k}. Then Δ_{X} is just an irreducible polynomial, which vanishes at f ∈ V^{∗} whenever the function f(x_{1}, …, x_{k}) has a multiple root, that is, vanishes at some v ∈ Cone(X), v ≠ 0, together with all first derivatives ∂f/∂x_{i}.

Consider now the d-dimensional projective space ℙ^{d} = ℙ(V) with homogeneous coordinates z_{0}, …, z_{d}, and let X ⊂ ℙ^{d} be the Veronese curve

(x^{d} : x^{d−1}y : x^{d−2}y^{2} : …: xy^{d−1} : y^{d}), x, y ∈ ℂ, (x, y) ≠ (0, 0)

(the image of the Veronese embedding ℙ^{1} ⊂ ℙ^{d}). Any linear form f(z) = ∑ a_{i}z_{i} is uniquely determined by its restriction to the cone Cone(X), which is a binary form f(x, y) = ∑ a_{i}x^{d−i}y^{i}. Therefore, f ∈ Cone(X^{∗}) if and only if f(x, y) vanishes at some point (x_{0}, y_{0}) ≠ (0, 0) (so (x_{0} : y_{0}) is a root of f(x, y)) with its first derivatives (so (x_{0} : y_{0}) is a multiple root of f(x, y)). It follows that Δ_{X} is the classical discriminant of a binary form.

1.2 Reflexivity Theorem

In this section we shall prove the Reflexivity Theorem. We follow the exposition in [GKZ2], which shows that there exists a deep connection between projective duality and symplectic geometry. Other proofs, including the investigation of a prime characteristic case, could be found, e.g., in [Se], [M], [Wa].

Theorem 1.7

(a) For any irreducible projective variety X ⊂ ℙ^{n}, we have X^{∗∗} = X.

(b) More precisely, If z is a smooth point of X and H is a smooth point of X^{∗}, then H is tangent to X at z if and only if z, regarded as a hyperplane in ℙ^{n}^{∗}, is tangent to X^{∗} at H.

The proof will be given in the next section.

1.2.A The Conormal Variety

We shall need some standard definitions.

Definition 1.8 If X is a smooth algebraic variety, then T X denotes the tangent bundle of X. If Y ⊂ X is a smooth algebraic subvariety then T Y is a subbundle in T X|_{Y}. The quotient T X|_{Y} /T Y is called the normal bundle of Y in X, denoted by N_{Y} X. By taking dual bundles we obtain the cotangent bundle T^{∗}X and the conormal bundle N_{Y}^{∗} X. The conormal bundle can be naturally regarded as a subvariety of T^{∗}X.

Recall that I_{X} ⊂ ℙ^{n} × ℙ^{n}^{∗} is the conormal variety and the dual variety X^{∗} coincides with pr_{2}(I_{X}). The projection pr_{1} : I_{X}^{0} → X_{sm} is a projective bundle, where pr_{1}, pr_{2} denote the projections of ℙ^{n} ×ℙ^{n}^{∗} to its factors. More precisely, pr_{1} identifies I_{X}^{0} with the projectivization ℙ(N_{X_{sm}}^{∗}ℙ^{n}) of the conormal bundle N_{X_{sm}}^{∗}ℙ^{n}. Indeed, the choice of a hyperplane H ⊂ ℙ^{n} tangent to X_{sm} at x is equivalent to the choice of a hyperplane T_{x}H in the tangent space T_{x}ℙ^{n}, which contains T_{x}X. The equation of this hyperplane is an element of N_{X_{sm}}^{∗}ℙ^{n} at x. The Reflexivity Theorem can be reformulated as follows:

I_{X} = I_{X^{∗}}.  (1.1)

It is more convenient to prove (1.1) by working with vector spaces instead of projective spaces.

We assume that ℙ^{n} = ℙ(V) and ℙ^{n}^{∗} = ℙ(V^{∗}). Then we have affine cones Y = Cone(X) ⊂ V and Y^{∗} = Cone(X^{∗}) ⊂ V^{∗}. We denote by Lag(Y) the closure of the conormal bundle N_{Y_{sm}}^{∗}V in the cotangent bundle T^{∗}V.

The space T^{∗}V is canonically identified with V × V^{∗}. Denote by pr_{1}, pr_{2} the projections of this product to its factors. Then Y^{∗} coincides with pr_{2}(Lag(Y)). Therefore, (1.1) can be reformulated as follows:

Lag(Y) = Lag(Y^{∗}),  (1.2)

where we identify T^{∗}V and T^{∗}V^{∗} with V × V^{∗}.

Recall that a smooth algebraic variety M is called the symplectic variety, if M admits a symplectic structure, that is, a differential 2-form ω with the following properties:

– ω is closed, dω = 0.

– ω is non-degenerate, for any p ∈ M the restriction of ω on T_{p}M is a nondegenerate skew product.

In this case dim M is necessarily even. An irreducible closed subvariety Λ ⊂ M is called Lagrangian if dim Λ = dim M/2 and the restriction of ω to the smooth locus Λ_{sm} vanishes as a 2-form (is totally isotropic).

The most important and the most typical (having in mind the Darboux Theorem) example of a symplectic variety is the cotangent bundle T^{∗}X of a smooth algebraic variety X. T^{∗}X carries a canonical symplectic structure defined as follows. Let (x_{1}, …, x_{n}) be a local coordinate system in X. Let ξ_{i} be the fiberwise linear function on T^{∗}X given by the pairing of a 1-form with the vector field ∂/∂x_{i}. Then (x_{1}, …, x_{n}, ξ_{1}, …, ξ_{n}) forms a local coordinate system in T^{∗}X. The form ω is defined by

ω = ∑_{i=1}^{n} dξ_{i} ∧ dx_{i}.  (1.3)

It easy to give an equivalent definition of ω without any coordinate systems. We shall define a canonical 1-form ν on T^{∗}X and then we shall take ω = dν. Let π : T^{∗}X → X be the canonical projection. Let p ∈ T^{∗}X and v ∈ T_{p}(T^{∗}X) be a vector tangent to T^{∗}X at p. Then ν(v) = p(π_{∗}v).

An important (and typical, as we shall see) example of a Lagrangian subvariety in T^{∗}X is obtained as follows. Let Y ⊂ X be any irreducible subvariety with smooth locus Y_{sm} and let

Lag(Y) = \overline{N_{Y_{sm}}^{∗}X}

be the closure of the conormal bundle of Z_{sm} (in T^{∗}X). Clearly, Lag(Y) is a conical subvariety (invariant under dilations of fibers of T^{∗}X).

Theorem 1.9

(a) Lag(Y) is a Lagrangian subvariety.

(b) Any conical Lagrangian subvariety has the form Lag(Y) for some irreducible subvariety Y ⊂ X.

Proof. Let us show that Lag(Y) is Lagrangian. Clearly, dim Lag(Y) = dim Y + (dim X − dim Y) = dim T^{∗}X/2. So we need only to verify that ω|_{Lag(Y)} = 0. It is sufficient to consider the smooth locus of Y only. Let x_{1}, …, x_{n} be a local coordinate system on X such that Y is locally defined by equations x_{1} = … = x_{r} = 0. Then the fibers of the conormal bundle over points of Y are generated by 1-forms dx_{1}, …, dx_{r}. Hence ξ_{r+1} = … = ξ_{n} = 0 on N_{Y_{sm}}^{∗}X and by (1.3) we see that ω = 0 on N_{Y_{sm}}^{∗}X. Therefore, Lag(Y) is indeed Lagrangian.

Suppose now that Λ ⊂ T^{∗}X is a conical Lagrangian subvariety. We take Y = pr(Λ), where pr : T^{∗}X → X is the projection, and claim that Λ = Lag(Y). It suffices to show that Λ ⊂ Lag(Y), because Λ and Lag(Y) are irreducible varieties of the same dimension. In turn, to prove that Λ ⊂ Lag(Y) it suffices to check that for any smooth point y ∈ Y the fiber pr^{−1}(y) ∩ Λ is contained in the conormal space (N_{Y}^{∗} X)_{y}. Let ξ be any covector from pr^{−1}(y) ∩ Λ. Since T_{y}^{∗}X is a vector space, we can regard ξ as a “vertical” tangent vector to T^{∗}X at a point y ⊂ X ⊂ T^{∗}X, where we identify X with the zero section of pr. Since Λ is conical, ξ ∈ T_{ξ}Λ. Therefore, since Λ is Lagrangian, ξ is orthogonal with respect to ω to any tangent vector from T_{ξ}Λ and, hence, to any tangent vector v ∈ T_{y}Y. But by (1.3) it is easy to see that this is equivalent to ξ ∈ (N_{Y}^{∗} X)_{y}. ⊓ ⊔

Now we can prove the Reflexivity Theorem. We shall prove (1.2). The identification T^{∗}V = V ×V^{∗} = T^{∗}V^{∗} takes the canonical symplectic structure on T^{∗}V to minus the canonical symplectic structure on T^{∗}V^{∗}. Therefore Lag(Y) regarded as a subvariety of T^{∗}V^{∗} is still Lagrangian. Moreover, clearly Lag(Y) ⊂ V ×V^{∗} is invariant under dilations of V and V^{∗}, therefore, Lag(Y) is a conical Lagrangian variety of T^{∗}V^{∗}. Therefore, by Theorem 1.9 Lag(Y) = Lag(Z), where Z is the projection of Lag(Y) on V^{∗}. But this projection coincides with Y^{∗}. Therefore, Lag(Y) = Lag(Y^{∗}). The Reflexivity Theorem is proved.

1.2.B Applications of the Reflexivity Theorem

Suppose that X ⊂ ℙ^{n} and X^{∗} ⊂ ℙ^{n}^{∗} are projectively dual varieties, I_{X} = I_{X^{∗}} ⊂ ℙ^{n} × ℙ^{n}^{∗} is the conormal variety. We have the diagram of projections

X ⟵^{pr_{1}} I_{X} ⟶^{pr_{2}} X^{∗}.

Theorem 1.10

(a) If X is smooth then I_{X} is smooth.

(b) If X^{∗} is a hypersurface then pr_{2} is a birational isomorphism.

(c) If X is smooth and X^{∗} is a hypersurface then pr_{2} is a resolution of singularities.

Proof. The map pr_{1} is a projective bundle over a smooth locus of X. Therefore, if X is smooth then I_{X} is smooth. If X^{∗} is a hypersurface then dim X^{∗} = dim I_{X} = n − 1. Since pr_{2} is generically a projective bundle, it is birational. Finally, (c) follows from (a) and (b). ⊓ ⊔

Typically the dual variety X^{∗} ⊂ ℙ^{n}^{∗} is a hypersurface. Namely, we shall see that if X^{∗} is not a hypersurface then X is ruled in projective subspaces.

Definition 1.11 We say that X is ruled in projective subspaces of dimension r if for any x ∈ X there exists a projective subspace L of dimension r such that x ∈ L ⊂ X. By a standard closedness argument it is sufficient to check this property only for points x from some Zariski open dense subset U ⊂ X.

Recall, that the number def X = codim_{ℙ^{n}}^{∗} X^{∗} − 1 is called the defect of X.

Theorem 1.12 Suppose that def X = r ≥ 1. Then

(1) X is ruled in projective subspaces of dimension r.

(2) If X is smooth then for any H ∈ X^{∗}_{sm} the contact locus Sing(H ∩ X) is a projective subspace of dimension r and the union of these projective subspaces is dense in X.

Proof. By Reflexivity Theorem 1.7, (a) is equivalent to the following: if codim X = r + 1 then X^{∗} is ruled in projective subspaces of dimension r. The condition for a hyperplane H to be tangent to X at a smooth point x is that T̂_{x}H contains T̂_{x}X. For a given x, all H with this property form a projective subspace of dimension r. But the set of hyperplanes of X^{∗} tangent to X at some smooth point obviously contains a Zariski open subset of X^{∗}.

(b) is proved by the same argument involving Reflexivity Theorem. ⊓ ⊔

Example 1.13 Suppose that X ⊂ ℙ^{n} is a non-linear curve. Then X^{∗} is a hypersurface. Indeed, X obviously could not contain a projective subspace ℙ^{k} for k > 0.

The following Theorem is also an easy corollary of Reflexivity Theorem. It allows to find singular points of hyperplane sections of smooth projective varieties.

Theorem 1.14 Suppose that X ⊂ ℙ^{n−1} is smooth and X^{∗} ⊂ ℙ^{n−1}^{∗} is a hypersurface. Let z_{1}, …, z_{n} be homogeneous coordinates on ℙ^{n−1} and a_{1}, …, a_{n} the dual homogeneous coordinates on ℙ^{n−1}^{∗}. Suppose that f = (a_{1}, …, a_{n}) is a smooth point of X^{∗}. Then the hyperplane section {f = 0} of X has a unique singular point with coordinates given by (\frac{∂Δ_{X}}{∂a_{1}} (f) : …: \frac{∂Δ_{X}}{∂a_{n}} (f)).

Proof. Let H ⊂ ℙ^{n−1} be the hyperplane corresponding to f. By the Reflexivity Theorem, H is tangent to X at z if and only if the hyperplane in ℙ^{n−1} corresponding to z is tangent to X^{∗} at f. Since X^{∗} is smooth at f, such a point z is unique and is given by z_{i} = \frac{∂Δ_{X}}{∂a_{i}} (f). ⊓ ⊔

1.3 Dual Plane Curves

1.3.A Parametric Representation of the Dual Plane Curve

Perhaps the most classical example of a dual variety is the dual curve C^{∗} of a non-linear plane curve C ⊂ ℙ^{2}. By definition, generic points of C^{∗} are the tangents to C at smooth points. In this case the Reflexivity Theorem has a fairly intuitive meaning. The tangent line T̂_{p} ∈ ℙ^{2}^{∗} at a smooth point p ∈ C is the limit of secants \overline{pq} for q ∈ C, q → p. Similarly, the point in ℙ^{2} that corresponds to the tangent to C^{∗} ⊂ ℙ^{2}^{∗} at a non-singular point T̂_{p} is the limit of the intersection points of the tangents T̂_{p} and T̂_{q} as q → p. Of course, this point is p.

In fact, it is quite easy to write down a parametric representation of C^{∗} using a given parametric representation of C. Let x, y, z be homogeneous coordinates on ℙ^{2} and p, q, r the dual homogeneous coordinates on ℙ^{2}^{∗}. We choose the affine chart ℂ^{2} = {z ≠ 0} ⊂ ℙ^{2} with affine coordinates x, y, so the third homogeneous coordinate z is set to be 1. The dual chart ℂ^{2}^{∗} ⊂ ℙ^{2}^{∗} with coordinates p, q is obtained by setting the third homogeneous coordinate r in ℙ^{2}^{∗} to be −1. Then ℂ^{2}^{∗} consists of lines in ℙ^{2} not passing through the point (0, 0) ∈ ℂ^{2} ⊂ ℙ^{2}. Every such line that meets ℂ^{2} is given by the affine equation px + qy = 1, the line in ℙ^{2} with coordinates p = q = 0 is the line “at infinity”. Suppose that a local parametric equation of C has the form x = x(t), y = y(t), where t is a local coordinate on C, and x(t), y(t) are analytic functions. By definition, the dual curve C^{∗} has the parametrization p = p(t), q = q(t), where p(t)x+q(t)y = 1 is the affine equation of the tangent line to C at the point (x(t), y(t)). Therefore, we have

p(t) = \frac{−y′(t)}{x′(t)y(t) − x(t)y′(t)}, q(t) = \frac{x′(t)}{x′(t)y(t) − x(t)y′(t)}.  (1.4)

Applying this formula two times we obtain Reflexivity Theorem once again.

Example 1.15 In analysis, there is a well-known duality between Banach spaces L_{p} and L_{q} for \frac{1}{p} + \frac{1}{q} = 1, see, e.g., [Ru]. Let us give an algebraic version of this duality.

Consider the curve X in ℂ^{2} given by

x^{a} + y^{a} = 1, a > 1, a ∈ ℚ.

This curve, and its closure in ℙ^{2}, is usually called the Fermat curve (especially if a ∈ ℤ). If a ∉ ℤ, this equation involves multivalued fractional power functions, but it is possible to put this equation into the polynomial form. In order to find the dual curve, we can use a parametric representation of X of the form

x = t, y = ^{a} \sqrt{1 − t^{a}}.

Using (1.4) we get the parametric representation of the dual curve as follows:

p = t^{a−1}, q = (1 − t^{a})^{\frac{a−1}{a}} .

The relation between p and q has the form

p^{b} + q^{b} = 1, where \frac{1}{a} + \frac{1}{b} = 1.

1.3.B The Legendre Transformation and Caustics

The projective duality is closely related to the Legendre transformation of classical mechanics, and, therefore, is in some sense analogous to the duality of the Lagrange and the Hamilton pictures of classical mechanics. To illustrate this analogy it will be sufficient to recall the classical definition of the Legendre transformation of real functions in one variable. Details can be found in [Ar].

Suppose that y = f(x) is a smooth convex real function, f′′(x) > 0. The Legendre transformation of the function f is a new function g of a new variable p, which is constructed in the following way. Consider the line y = px. We take the point x = x(p) at which the graph of y = f(x) has a slope p (so f′(x(p)) = p), and define g(p) as g(p) = px(p) − f(x(p)). Equivalently, we define x(p) as a unique point, where the function F (p, x) = px − f(x) has a maximum with respect to x and define g(p) = F (p, x(p)).

The Legendre transformation is easily seen to be involutive. To see how it is related to the dual curve, let us notice that the straight line y = G(x, p) = xp − g(p) is nothing else but a tangent line to the graph of f with slope p.

To link projective duality and Legendre transformation we need a notion of a caustic curve. To introduce it, let us express the projective duality entirely in terms of the projective plane ℙ^{2}. By definition, a tangent line to a curve C at some point x is the line that contains x and which is infinitesimally close to the curve C near x. A point of ℙ^{2}^{∗} is a line l ⊂ ℙ^{2}. A curve in ℙ^{2}^{∗} is a 1-parameter family of lines in ℙ^{2}. For example, a line in ℙ^{2}^{∗} is a pencil x^{∗} of all lines in ℙ^{2} passing through a given point x ∈ ℙ^{2}. The dual curve C^{∗} is a 1-parameter family of tangent lines to C. Suppose now that C′ ⊂ ℙ^{2}^{∗} is some curve (some 1-parameter family of lines in ℙ^{2}). Let us find a geometric interpretation of the dual curve C′^{∗} ⊂ ℙ^{2}. Take some line l ∈ C′. The condition that x^{∗} is tangent to C′ at l means that the line l ∈ ℙ^{2}^{∗} is a member of a family C′ and other lines from C′ near l are infinitesimally close to the pencil of lines x^{∗}. This is usually expressed by saying that x is a caustic point for the family of lines C′. The set of all caustic points of the family of lines C′ is usually called the caustic (or the envelope) of C′. This is nothing else but the projectively dual curve C′^{∗}. Now the Reflexivity Theorem means that any curve C ⊂ ℙ^{2} coincides with the caustic curve for the family of its tangent lines. The “dual” form of this theorem is less intuitively obvious, it means that any 1-parameter family of lines in ℙ^{2} consists of tangent lines to some curve C and this curve is a caustic curve for this family of lines. For (real) families of lines the caustic could be found (locally) with the help of the Legendre transformation:

Theorem 1.16 Consider a family of real straight lines y = px − g(p). Then its caustic curve has the equation y = f(x), where f is the Legendre transformation of g.

This Theorem is, in fact, a version of the Reflexivity Theorem, see [Ar] for the proof.

1.3.C Correspondence of Branches. Plücker Formulas.

Even if a plane curve C ⊂ ℙ^{2} is smooth, the dual curve C^{∗} ⊂ ℙ^{2}^{∗} almost always has singularities. We have a natural map C → C^{∗}, sending a point p ∈ C to a tangent line l to C at p. This map is clearly a resolution of singularities. In general, curves C and C^{∗} are birationally equivalent. Indeed, consider the conormal variety I_{C} ⊂ ℙ^{2} × ℙ^{2}^{∗}, the closure of the set of pairs (p, l), p ∈ C_{sm}, l ∈ C^{∗}_{sm}, l is tangent to C at p. Then, by Reflexivity Theorem, I projects birationally both on C and C^{∗}. Therefore, C and C^{∗} are birationally equivalent, in particular, they have the same geometric genus g.

A line l which is tangent to C in at least two points, is a singular point of C^{∗}. It is known as the multiple tangent. If a multiple tangent l has exactly two tangency points on C and the intersection multiplicity at each of them is equal exactly to 2, then l is called the bitangent. A bitangent corresponds to an ordinary double point of C^{∗}.

If the tangent l = T̂_{p} at a non-singular point p ∈ C intersects C at p with multiplicity ≥ 3, it is again a singular point on C^{∗}. If the intersection multiplicity is precisely 3, and l is not tangent to C at any other point, then p is called an inflection point (or flex) of C. Then l is a cuspidal point (or cusp) of C^{∗}.

Now we may introduce a class of “generic” curves with singularities, which is preserved by the projective duality. Namely, we say that a curve C is generic if both C and C^{∗} have only double points and cusps as their singularities. Suppose that C is generic in this sense. Let d, g, κ, δ, b, f be the degree, the geometric genus, the number of cusps, the number of double points, the number of bitangents, and the number of flexes of C. Let d^{∗}, g^{∗}, κ^{∗}, δ^{∗}, b^{∗}, f^{∗} be the corresponding numbers for C^{∗} (d^{∗} is also sometimes called the class of C). Then by Reflexivity Theorem we have the following

Proposition 1.17 g = g^{∗}, κ = f^{∗}, δ = b^{∗}, b = δ^{∗}, f = κ^{∗}.

It turns out that there is another remarkable set of equations linking these numbers, that was discovered by Plücker and Clebsch. The proof can be found in [GH].

Theorem 1.18

g = \frac{1}{2}(d^{∗} − 1)(d^{∗} − 2) − b − f,

g = \frac{1}{2}(d − 1)(d − 2) − δ − κ,

d = d^{∗}(d^{∗} − 1) − 2b − 3f,

d^{∗} = d(d − 1) − 2δ − 3κ.

Example 1.19 Let C ⊂ ℙ^{2} be a smooth conic. In homogeneous coordinates x_{1}, x_{2}, x_{3}, the curve C is given by

(Ax, x) = ∑_{i,j=1}^{3} a_{ij}x_{i}x_{j} = 0,

where A = ||a_{ij}|| is a non-degenerate symmetric 3 × 3-matrix. It is clear that the tangent line to C at a point x_{0} ∈ C is given by the equation (Ax_{0}, x) = 0. Hence the point ξ ∈ ℙ^{2}^{∗} corresponding to this tangent line has homogeneous coordinates Ax_{0}, which implies (A^{−1}ξ, ξ) = 0. Therefore, C^{∗} ⊂ ℙ^{2}^{∗} is also a smooth conic defined by the inverse matrix A^{−1}.

Example 1.20 Let C ⊂ ℙ^{2} be a smooth cubic curve. By Bezout Theorem, C is automatically generic and Plücker formulas are always applicable. C has no bitangents and has exactly 9 flexes. The dual curve C^{∗} is a very special curve of degree 6 with 9 cusps and no double points. Schläffli has found a beautiful determinantal formula for C^{∗}. Let x_{1}, x_{2}, x_{3} be homogeneous coordinates in ℙ^{2} and p_{1}, p_{2}, p_{3} the dual coordinates in ℙ^{2}^{∗}. Let f(x_{1}, x_{2}, x_{3}) = 0 be the homogeneous equation of C and F (p_{1}, p_{2}, p_{3}) = 0 be the homogeneous equation of C^{∗}. Consider the polynomial

V (p, x) = | \begin{matrix}0 & p_{1} & p_{2} & p_{3} \\ p_{1} & \frac{∂^{2}f}{∂x_{1}∂x_{1}} & \frac{∂^{2}f}{∂x_{1}∂x_{2}} & \frac{∂^{2}f}{∂x_{1}∂x_{3}} \\ p_{2} & \frac{∂^{2}f}{∂x_{2}∂x_{1}} & \frac{∂^{2}f}{∂x_{2}∂x_{2}} & \frac{∂^{2}f}{∂x_{2}∂x_{3}} \\ p_{3} & \frac{∂^{2}f}{∂x_{3}∂x_{1}} & \frac{∂^{2}f}{∂x_{3}∂x_{2}} & \frac{∂^{2}f}{∂x_{3}∂x_{3}} \end{matrix} | .

Clearly V (p, x) has degree 2 in x. Then Schläffli’s formula is as follows:

Theorem 1.21

F (p_{1}, p_{2}, p_{3}) = | \begin{matrix}0 & p_{1} & p_{2} & p_{3} \\ p_{1} & \frac{∂^{2}V}{∂x_{1}∂x_{1}} & \frac{∂^{2}V}{∂x_{1}∂x_{2}} & \frac{∂^{2}V}{∂x_{1}∂x_{3}} \\ p_{2} & \frac{∂^{2}V}{∂x_{2}∂x_{1}} & \frac{∂^{2}V}{∂x_{2}∂x_{2}} & \frac{∂^{2}V}{∂x_{2}∂x_{3}} \\ p_{3} & \frac{∂^{2}V}{∂x_{3}∂x_{1}} & \frac{∂^{2}V}{∂x_{3}∂x_{2}} & \frac{∂^{2}V}{∂x_{3}∂x_{3}} \end{matrix} | .

The proof can be found in [GKZ2].

Example 1.22 Suppose that C ⊂ ℙ^{2} is a generic smooth quartic curve. By applying Plücker formulas we see that C has genus 3, 24 flexes, and 28 bitangents. There are two known visual descriptions of these bitangents. The first classical approach is to realize the quartic curve as a ‘shade’ of a cubic surface in ℙ^{3}. Namely, suppose that S ⊂ ℙ^{3} is a generic cubic surface, x ∈ S is a generic point. Consider the projective plane ℙ^{2} of lines in ℙ^{3} passing through x. Then lines l ∈ ℙ^{2} that are tangent to S form a quartic curve C. The 28 bitangents then coincide with projections of 27 lines on S plus one extra line: the projectivization of a tangent space T_{x}S.

Another classical description is less known but it is more in style of these notes. Consider a generic 3-dimensional linear system L of quadrics in ℙ^{3} (an element of Gr(3, S^{2}(ℂ^{4})^{∗}). Then singular members of this linear system form a quartic curve in ℙ(L) = ℙ^{2}. It can be shown that any generic quartic curve can be obtained this way (for example by calculating the differential at a generic point of the map from linear systems to quartic forms). By the Bezout theorem L has 2^{3} = 8 base points in ℙ^{3}. For any two base points p, q let l_{pq} ⊂ ℙ(L) be the line formed by all quadrics that contain not only points p and q, but also a line [pq] connecting them. Therefore, we have 28 lines l_{pq} ⊂ ℙ(L). Let us show that these lines are bitangents to C.

Fix a basis {e_{1}, e_{2}, e_{3}, e_{4}} in ℂ^{4} such that e_{1} belongs to p and e_{2} belongs to q. Then quadrics from l_{pq} have the form

G = ( \begin{matrix}0 & A^{T} \\ A & B \end{matrix} ) ,

therefore,

det G = (det A)^{2}.

This exactly means that l_{pq} intersects C in two double points, i.e. l_{pq} is a bitangent to C.

1.4 Projections and Linear Normality

1.4.A Projections

Let P = ℙ(V) be an n-dimensional projective space and L ⊂ P a projective subspace of dimension k, L = ℙ(U), U ⊂ V. The quotient projective space P/L = ℙ(V/U) has, as points, (k + 1)-dimensional projective subspaces in P containing L. The projection with center L is the map π_{L} : P \ L → P/L which takes any point x ∈ P \ L to the (k + 1)-dimensional projective subspace spanned by x and L. In other words, it corresponds to the projection V → V/U. Classically, both P/L and π_{L} can be visualized inside the initial projective space P. Namely, P/L can be identified with any (n − k − 1)- dimensional projective subspace H ⊂ P not intersecting L. Then π_{L} sends any point x ∈ P \ L to the unique intersection point of H and (k + 1)- dimensional projective subspace spanned by x and L. Clearly, the dual projective space (P/L)^{∗} is canonically embedded in P^{∗} as the set of hyperplanes containing L, so it coincides with L^{∗}, the dual variety of L.

Theorem 1.23

(a) Let X ⊂ P = ℙ(V) be an algebraic subvariety not intersecting the projective subspace L = ℙ(U) and such that dim X < dim P/L. Then

(π_{L}(X)^{∗} ⊂ L^{∗} ∩ X^{∗}.

The discriminant Δ_{π_{L}(X)} (as a polynomial on (V/U)^{∗} ⊂ V^{∗}) is a factor of the restriction of Δ_{X} to (V/U)^{∗}.

(b) Suppose further that π_{L} : X → π_{L}(X) is an isomorphism. Then

(π_{L}(X)^{∗} = L^{∗} ∩ X^{∗}, and Δ_{π_{L}(X)} = Δ_{X}|_{(V/U)^{∗}}.

Proof. A hyperplane in P/L is just a hyperplane in P containing L. It is clear that if a hyperplane H ⊂ P/L is tangent to π_{L}(X) at some smooth point y = π_{L}(x), where x ∈ X is also smooth, then H as a hyperplane in P is tangent to X at x. This proves (a).

Suppose now that π_{L} is an isomorphism. The same argument as above also shows that if X_{0}^{∗} ⊂ X^{∗} (resp. π_{L}(X)_{0}^{∗} ⊂ π_{L}(X)^{∗}) is a dense subset of hyperplanes tangent to X (resp. π_{L}(X)) at some smooth point then π_{L}(X)_{0}^{∗} = L^{∗} ∩ X_{0}^{∗}, since π_{L} induces an isomorphism of smooth loci X_{sm} → π_{L}(X)_{sm}. Moreover, it is clear that at a generic point of π_{L}(X)_{0}^{∗} varieties L^{∗} and X_{0}^{∗} intersect transversally. Since X^{∗} = \overline{X_{0}^{∗}} and π_{L}(X)^{∗} = \overline{π_{L}(X)_{0}^{∗}}, in order to prove (b) it suffices to show that L^{∗} ∩ X^{∗} = \overline{L^{∗} ∩ X_{0}^{∗}}. Here the assumption that π_{L} is an isomorphism is crucial, because otherwise critical points of π_{L} may produce extra components of L^{∗} ∩ X^{∗}.

We need to show that if H(t) is a 1-parameter family of hyperplanes such that for t ≠ 0 the hyperplane H(t) is tangent to X at some smooth point x(t) and H(0) contains L, then there exists another 1-parameter family H′(t) such that H′(0) = H(0) and for t ≠ 0 the hyperplane H′(t) is tangent to X at x(t) and contains L. Since X is projective, hence compact, we can suppose that the limit x(0) = lim_{t→0} x(t) exists. However, x(0) may be a singular point of X. Let T̂_{x(0)}X be an embedded Zariski projective tangent space to X at x(0). Since π_{L} is an isomorphism, it induces an isomorphism of Zariski tangent spaces. Therefore T̂_{x(0)}X does not intersect L. Let T_{0} ⊂ T̂_{x(0)}X be a limit position of embedded tangent Zariski spaces T_{t} = T̂_{x(t)}X as t → 0. Then for any t the projective subspace T_{t} does not intersect L. Therefore we may consider a family T′_{t} of (dim L + dim X + 1)-dimensional projective subspaces such that for any t we have L ⊂ T′_{t} and for t ≠ 0 the subspace T′_{t} is tangent to X at x(t). Namely, T′_{t} is a projective subspace spanned by L and T_{t}. Since dim X < dim P/L, we can embed T′_{t} into a family of hyperplanes H′_{t} with same properties. ⊓ ⊔ 1.4.B Linear Normality

Definition 1.24 The projective variety X ⊂ ℙ^{n} is called non-degenerate if X is not contained in any hyperplane H.

The next theorem shows that we may restrict ourselves to non-degenerate varieties while studying dual varieties and discriminants. First we shall need the following definition. We recall from a previous section that if ℙ^{k} = L ⊂ ℙ^{n} is a projective subspace then a dual projective space ℙ^{k}^{∗} is not canonically embedded in ℙ^{n}^{∗}, but (by some abuse of notation) is canonically isomorphic to the quotient projective space ℙ^{n}^{∗}/L^{∗}, where L^{∗} is a projective subspace of ℙ^{n}^{∗} projectively dual to L. However, if Y ⊂ ℙ^{k}^{∗} is any subvariety then we can canonically define a cone over Y with vertex L^{∗} as the closure π_{L^{∗}}^{−1}(Y).

Theorem 1.25 Let X ⊂ ℙ^{n} be an irreducible subvariety.

(a) Assume that X is contained in a hyperplane H = ℙ^{n−1}. If X^{∗}′ is the dual variety of X, when we consider X as a subvariety of ℙ^{n−1}, then X^{∗} is the cone over X^{∗}′ with vertex p corresponding to H.

(2) Conversely, if X^{∗} is a cone with vertex p, then X is contained in the corresponding hyperplane H.

Proof. If H′ ≠ H is a tangent hyperplane of X then H ∩ H′ is a tangent hyperplane of X in ℙ^{n−1}. Conversely, if T is a tangent hyperplane of X in ℙ^{n−1} then each hyperplane H′ in ℙ^{n} containing T is tangent to X. Therefore, X^{∗} is the cone over X^{∗}′. This proves (a).

By Reflexivity Theorem, we also have (b). Namely, each hyperplane which is tangent to X^{∗} at a smooth point necessarily contains p. Therefore X = X^{∗∗} is contained in the hyperplane corresponding to p. ⊓ ⊔

In terms of discriminants, Theorem 1.25 can be reformulated as follows. Consider a surjection π : V → U of vector spaces and consider ℙ(V^{∗}), ℙ(U^{∗}). Then we have an embedding i : ℙ(U^{∗}) ↪ ℙ(V^{∗}). Let X ⊂ ℙ(U^{∗}). Then Δ_{X} is a polynomial function on U. If we consider X as a subvariety in ℙ(V^{∗}) then Δ_{i(X)} is a function on V. By Theorem 1.25 these polynomial functions are related as follows:

Δ_{i(X)}(f) = Δ_{X}(π(f)).

In other words, Δ_{i(X)} does not depend on some of the arguments and forgetting these arguments gives Δ_{X}.

Theorems 1.23 and 1.25 show that in order to study dual varieties and discriminants we need to consider only projective varieties X that are nondegenerate and not equal to a non-trivial projection. These projective varieties are called linearly normal. To give a more intrinsic definition of linearly normal varieties we need to recall the correspondence between invertible sheaves, linear systems, and projective embeddings.

An invertible sheaf on an irreducible algebraic variety X is simply the sheaf of sections of some algebraic line bundle. For example, the structure sheaf of regular functions O_{X} corresponds to the trivial line bundle. Usually we shall not distinguish notationally between invertible sheaves and line bundles. Invertible sheaves form the group Pic(X) with respect to the tensor product.

A Cartier divisor on X is a family (U_{i}, g_{i}), i ∈ I, where U_{i} are open subsets of X covering X, and g_{i} are rational functions on U_{i} such that g_{i}/g_{j} is regular on each intersection U_{i} ∩ U_{j}. The functions g_{i} are called local equations of the divisor. More precisely, a Cartier divisor is an equivalence class of such data. Two collections (U_{i}, g_{i}) and (U′_{i}, g′_{i}) are equivalent if their union is still a divisor. Cartier divisors can be added, by multiplying their local equations. Thus they form a group, denoted by Div(X).

If each local equation g_{i} is regular on U_{i}, then we say that the divisor D is effective, and we write D ≥ 0. The subschemes {g_{i} = 0} of the U_{i} can then be glued together into a subscheme of X, also denoted by D. Therefore, effective Cartier divisors can be identified with subschemes of X that are locally given by one equation. Any non-zero rational function f ∈ ℂ(X) determines a Cartier divisor (f) which is said to be principal. Principal divisors form a subgroup of Div(X).

Let K_{X} denote the sheaf of rational functions on X, K_{X}(U) = ℂ(U). To every Cartier divisor D = (U_{i}, g_{i})_{i∈I} we can attach a subsheaf O_{X}(D) ⊂ K_{X}. Namely, on U_{i} it is defined as g_{i}^{−1}O_{U_{i}}. On the intersection U_{i} ∩U_{j} the sheaves g_{i}^{−1}O_{U_{i}} and g_{j}^{−1}O_{U_{j}} coincide, since g_{i}/g_{j} is invertible. Therefore these sheaves can be pasted together into a sheaf O_{X}(D) ⊂ K_{X}. For instance, O_{X}(0) = O_{X} and O_{X}(D_{1} + D_{2}) = O_{X}(D_{1}) ⊗ O_{X}(D_{2}). The sheaves O_{X}(D) are invertible. In fact, multiplication by g_{i} defines an isomorphism (“trivialization”) O_{X}(D)|_{U_{i}} ≃ O_{U_{i}}. Therefore, we have a homomorphism Div(X) → Pic(X). This homomorphism is surjective and its kernel consists of principal divisors.

A non-zero section of O_{X}(D) is a rational function f on X such that the functions fg_{i} are regular on the U_{i}, in other words, such that the divisor (f) + D is effective. If D itself is effective, the sheaf O_{X}(D) has a canonical section s_{D}, which corresponds to the constant function 1. By contrast, the sheaf O_{X}(−D), for D effective, is an ideal sheaf in O_{X}. It defines D as a subscheme. The sections of invertible sheaves define some divisors. Let s ∈ H^{0}(X, L) be a non-trivial global section of an invertible sheaf L. After choosing some trivializations ϕ_{i} : L_{U_{i}} ≃ O_{U_{i}} on a covering (U_{i}), we obtain an effective divisor (U_{i}, ϕ_{i}(s_{i})), which we denote by div(s, L). For instance, if D is effective, the canonical section s_{D} of the sheaf O(D) defines D. Thus we have established the possibility to define any effective divisor by one equation s = 0, keeping in mind that s is not a function, but a section of an invertible sheaf. If s′ is another non-zero global section of L then the divisors div(s′, L) and div(s, L) differ by the divisor of a rational function s/s′. One also says that they are linearly equivalent.

Example 1.26 Let us recall the construction of invertible sheaves on projective spaces ℙ(V). All these sheaves have the form O(d), where O(d) is the sheaf of degree d homogeneous functions on ℙ(V). More precisely, let π : V \ {0} → ℙ(V) be the canonical projection. If U ⊂ ℙ(V) is a Zariski open set, then the sections of O(d) over U are, by definition, regular functions f on π^{−1}(U) ⊂ V, which are homogeneous of degree d, f(λv) = λ^{d}f(v). It is well-known that these sheaves are invertible and any invertible sheaf has the form O(d) for some d. For example, O(−1), as a line bundle, is the tautological line bundle, the fiber of O(−1), as a line bundle, over a point of ℙ(V) represented by a 1-dimensional subspace l is l itself. If d < 0 then H^{0}(ℙ, O(d)) = 0. If d ≥ 0 then H^{0}(ℙ, O(d)) = S^{d}V^{∗}, homogeneous polynomials of degree d. For any non-zero s ∈ H^{0}(ℙ, O(d)) the corresponding effective divisor div(s, O(d)) is just the hypersurface defined by the polynomial s. In particular, hyperplanes in ℙ correspond to global sections of O(1).

Suppose now that X is a projective variety in ℙ^{n}. Then any sheaf O(d) can be restricted on X, thus we get a sheaf O_{X}(d) for any d. In particular, we have a restriction homomorphism of global sections:

V^{∗} = H^{0}(ℙ^{n}, O(1)) →^{res} H^{0}(X, O_{X}(1)).

Clearly this map is not injective if and only if X is degenerate. So suppose now that X is non-degenerate. In general, map res is not surjective. Its image is a vector subspace W ⊂ H^{0}(X, O_{X}(1)) with the following obvious property: for any x ∈ X there exist some section s ∈ W such that s(x) ≠ 0. Clearly divisors of the form div(s, O_{X}(1)), s ∈ W are just hyperplane sections X ∩H for various hyperplanes H.

In general, if X is any variety with invertible sheaf L then any family of divisors |W | of the form div(s, Λ), where s ∈ W ⊂ H^{0}(X, L), dim W < ∞, is called a linear systems of divisors. If X is projective (or just complete) then H^{0}(X, L) is finite-dimensional and we can take W = H^{0}(X, L). The corresponding linear system is called complete. It is denoted by |L|, or by |D| if L = O(D). Every effective divisor that is linearly equivalent to D appears in |D|, and exactly once.

A linear system W is called base-point free if the intersection of all its divisors is empty, or, equivalently, if for any x ∈ X there exists some s ∈ W such that s(x) ≠ 0. As we have seen, if X ⊂ ℙ(V) is a projective variety then the image of res : V^{∗} → H^{0}(X, O_{X}(1)) defines a base-point free linear system. It turns out that any base-point free linear system |W | defines a morphism into a projective space. Namely, for any x ∈ X we have a hyperplane Γ (x) ⊂ W consisting of all sections vanishing at x. Therefore, we have a regular map Γ : X → ℙ(W^{∗}). If Γ is an embedding then we see that W is identified with the image of a restriction map W → H^{0}(X, O_{X}(1)) and O_{X}(1) is identified with L. So, we need a final definition.

Definition 1.27 A linear system |W | is called very ample if it defines an embedding in a projective space. An invertible sheaf L is called very ample if the complete linear system |L| is very ample. L is called ample if some tensor power L^{⊗m}, m > 1, is very ample.

In particular, we see that non-degenerate projective embeddings of X are in a 1 − 1 correspondence with very ample linear systems of divisors.

Now we can describe linearly normal varieties.

Theorem 1.28 A projective variety X ⊂ ℙ(V) is linearly normal if and only if the restriction homomorphism res : V^{∗} → H^{0}(X, O_{X}(1)) is an isomorphism if and only if the embedding X ⊂ ℙ(V) is given by a complete linear system corresponding to some very ample invertible sheaf L on X.

Proof. Clearly res is injective if and only if X is non-degenerate. Suppose that res is not surjective. Then the linear system |O_{X}(1)| gives a non-degenerate embedding of X to a higher-dimensional vector space H^{0}(X, O_{X}(1)). The initial embedding is obtained from this one by the projection with center ℙ(Ann(Im(res))). Conversely, if X can be obtained as a non-trivial isomorphic projection of a non-degenerate variety X̃ in a larger projective space ℙ(U) then we have a proper embedding V^{∗} ⊂ U^{∗}. The map U^{∗} → H^{0}(X, OX̃(1)), given by restricting linear functions from ℙ(U) to X̃, is an injection, since X̃ is non-degenerate. Thus the map res is not surjective, being the composition V^{∗} ↪ U^{∗} → H^{0}(X, O_{X}(1)). ⊓ ⊔

We see that in order to study dual varieties and discriminants we can restrict ourselves to linearly normal varieties.

Definition 1.29 A pair (X, L) of a projective variety and a very ample invertible sheaf on it is called the polarized variety.

Any polarized variety admits a canonical embedding in a projective space with a linearly normal image, namely, the embedding given by the complete linear system |L|. Therefore we may speak about dual varieties, defect, discriminants, etc. of polarized varieties without any confusion.

1.5 Dual Varieties of Smooth Divisors

Suppose that X ⊂ ℙ^{N} is a smooth projective variety. The following theorem allows to find the defect of smooth hyperplane or hypersurface sections of X.

Theorem 1.30 ([E2, HK, Ho1])

(a) Assume that Y = X ∩ H is a smooth hyperplane section of X. Then

def Y = max{0, def X − 1}.

Moreover, if X^{∗} is not a hypersurface, then the dual variety Y^{∗} is the cone over X^{∗} with H as a vertex.

(b) Assume that Y is a smooth divisor corresponding to a section of O_{X}(d) for d ≥ 2. Then def Y = 0.

Proof. Consider the conormal variety

I_{X} = ℙ(N_{X}^{∗}ℙ^{N}(1)) ⊂ ℙ^{N} × ℙ^{N}^{∗}

and the following diagram of projections

ℙ^{N} ⊃ X ←^{π_{1}} I_{X} →^{π_{2}} X^{∗} ⊂ ℙ^{N}^{∗}.

We use notation O_{I_{X}} (a, b) for the line bundle π_{1}^{∗}O_{X}(a) ⊗ π_{2}^{∗}O_{X^{∗}}(b). We will need the notion of the tautological line bundle on a projective bundle. Let E be a vector bundle on X. We consider the variety X_{E} = ℙ(E^{∗}), the projectivization of the bundle E^{∗}. There is a projection p : E_{X} → X whose fibers are projectivizations of fibers of E^{∗}, and a projection π : E^{∗}\X → X_{E}, where X is embedded into the total space of E^{∗} as the zero section. We denote by ξ(E) the tautological line bundle on X_{E} defined as follows. For open U ⊂ X_{E}, a section of ξ(E) over U is a regular function on π^{−1}(U) which is homogeneous of degree 1 with respect to dilations of E^{∗}. The restriction of ξ(E) to every fiber p^{−1}(x) = ℙ(E_{x}^{∗}) is the tautological line bundle O(1) of the projective space ℙ(E_{x}^{∗}). For example, O_{I_{X}} (0, 1) is the tautological line bundle of ℙ(N_{X}^{∗}ℙ^{N}(1)).

Suppose that Y is a smooth member of a linear system |O_{X}(d)|. Clearly we have N_{Y} X = O_{Y} (d). Consider the following exact sequence

0 → O_{Y} (d − 1) → N_{Y} ℙ^{N}(−1) → N_{X}ℙ^{N}(−1)|_{Y} → 0.

Let D = ℙ(N_{X}^{∗}ℙ^{N}(1)|_{Y}). Then there is a natural embedding of D as a divisor in I_{Y} = ℙ(N_{Y}^{∗} ℙ^{N}(1)). We see that O_{I_{Y}} (D) ≃ O_{I_{Y}} (1 − d, 1).

We shall study the following diagram

ℙ^{N}^{∗} ⊃ Y^{∗} ⊃ π_{2}(D) ⊂ X^{∗} ⊂ ℙ^{N}^{∗}
↑^{π_{2}}  ↑^{π_{2}}  ↑^{π_{2}}
I_{Y} ⊃ D ⊂ I_{X}
↓^{π_{1}}  ↓^{π_{1}}  ↓^{π_{1}}
ℙ^{N} ⊃ Y = Y ⊂ X ⊂ ℙ^{N}

Consider the morphism g given by the complete linear system |O_{I_{Y}} (0, 1)|. Since π_{2}^{∗}O_{Y^{∗}}(1) = O_{I_{Y}} (0, 1), we can regard Y^{∗} as the linear projection of g(I_{Y}). Thus dim Y^{∗} = dim g(I_{Y}).

(a) Assume that d = 1. Then O_{I_{Y}} (0, 1) = O_{I_{Y}} (D). Therefore g(D) is a hyperplane section of g(I_{Y}) and

dim g(D) = dim π_{2}(D) = dim Y^{∗} − 1.

Therefore it is sufficient to prove that (A) If def X = k > 0, then dim π_{2}(D) = dim X^{∗}. (B) If def X = 0 then dim π_{2}(D) = dim X^{∗} − 1.

For any q ∈ X^{∗} the preimage π_{2}^{−1}(q) is isomorphic to the contact locus of the hyperplane q = 0 and X. Therefore, dim π_{2}^{−1}(q) ≥ def X and for generic q ∈ X^{∗} we have an equality. Therefore, if def X = k > 0 then dim D ∩ π_{2}^{−1}(q) > 0 and (A) follows. Moreover, we see that any hyperplane tangent to X is also tangent to Y, therefore, Y^{∗} is a cone over X^{∗} with the vertex H. Suppose now that def X = 0. Then π_{2} is birational, and, by the same argument as above, dim π_{2}(D) ≤ dim X^{∗} − 1. Moreover, if Y passes through a point x ∈ X such that there exists a hyperplane H ∈ X^{∗}_{sm} tangent to X at x then, in fact, dim π_{2}(D) = dim X^{∗} − 1. This proves (B) for generic hyperplane sections and, hence, for arbitrary smooth hyperplane sections by a standard argument.

(b) Assume d ≥ 2. We claim that g|_{I_{Y} \D} is one to one, and, therefore, dim Y^{∗} = dim g(I_{Y}) = dim I_{Y} = N − 1 and Y^{∗} is a hypersurface. Indeed, let y_{1} and y_{2} be two distinct points in I_{Y} \ D. We need to show that the linear system |O_{I_{Y}} (0, 1)| separates y_{1} and y_{2}. If π_{1}(y) = π_{1}(y_{2}) then π_{2}(y_{1}) ≠ π_{2}(y_{2}) because π_{2} maps π_{1}^{−1}(π_{1}(y_{1})) isomorphically to a projective subspace in Y^{∗}. So we may assume that π_{1}(y_{1}) ≠ π_{1}(y_{2}). Then we can find a hypersurface S of degree d − 1 in ℙ^{N} which will contain π_{1}(y_{1}) but not π_{1}(y_{2}). Recall that O_{I_{Y}} (D) ≃ O_{I_{Y}} (1 − d, 1). Therefore, D + π_{1}^{−1}(S) ∈ |O_{I_{Y}} (0, 1)|. But y_{1} ∈ D + π_{1}^{−1}(S) and y_{2} ∉ D + π_{1}^{−1}(S). ⊓ ⊔

20 1. Dual Varieties

2. Dual Varieties of Algebraic Group Orbits

Preliminaries

It is quite difficult to find the explanation of the following phenomenon: most interesting discriminants and dual varieties appear when the projective variety X is either homogeneous or a closure of a homogeneous variety. There are three known big classes of examples: projective embeddings of flag varieties, algebraic groups acting on projective spaces with finitely many orbits, and projective embeddings of torical varieties. The last case leads to the theory of A-discriminants studied in detail in the beautiful book [GKZ2] and omitted here.

2.1 Polarized Flag Varieties

2.1.A Definitions and Notations

Let G be a connected simply-connected semisimple complex algebraic group with a Borel subgroup B and a maximal torus T ⊂ B. Let P be the character group of T (the weight lattice). Let Δ ⊂ P be the set of roots of G relative to T. To every root α ∈ Δ we can assign the 1-dimensional unipotent subgroup U_{α} ⊂ G. We define the negative roots Δ^{−} as those roots α such that U_{α} ⊂ B. The positive roots are Δ^{+} = Δ \ Δ^{−} = −Δ^{−}. Let Π ⊂ Δ^{+} be simple roots, Π = {α_{1}, …, α_{n}}, where n = rank G = dim T. Any root α ∈ Δ is an integral combination ∑_{i} n_{i}α_{i} with nonnegative n_{i} (for α ∈ Δ^{+}) or nonpositive n_{i} (for α ∈ Δ^{−}). If G is simple then we use the Bourbaki numbering [Bo] of simple roots.

The weight lattice P is generated as a ℤ-module by the fundamental weights ω_{1}, …, ω_{n} dual to the simple roots under the Killing form

⟨ω_{i}|α_{j}⟩ = \frac{2(ω_{i}, α_{j})}{(α_{j}, α_{j})} = δ_{ij}.

A weight λ = ∑_{i} n_{i}ω_{i} is called dominant if all n_{i} ≥ 0. We denote the dominant weights by P^{+}. Dominant weights parametrize finite-dimensional irreducible G-modules: to any λ ∈ P^{+} we assign an irreducible G-module V_{λ} with highest weight λ. A weight λ = ∑_{i} n_{i}ω_{i} is called strictly dominant if all n_{i} > 0. We denote the strictly dominant weights by P^{++}. There is a partial order on P: λ > μ if λ − μ is an integral combination of simple roots with nonnegative coefficients. If λ ∈ P^{+} then we denote by λ^{∗} ∈ P^{+} the highest weight of the dual G-module V_{λ}^{∗}. Let W be the Weil group of G relative to T. If w_{0} ∈ W is the longest element then λ^{∗} = −w_{0}(λ).

The character group of B is identified with a character group of T. Therefore, for any λ ∈ P we can assign a 1-dimensional B-module ℂ_{λ}, where B acts on ℂ_{λ} by a character λ. Now we can define the twisted product G×_{B} ℂ_{λ} to be the quotient of G × ℂ_{λ} by the diagonal action B:

b · (g, z) = (gb^{−1}, λ(b)z).

Projection onto the first factor induces the map G ×_{B} ℂ_{λ} → G/B, which realizes the twisted product as an equivariant line bundle L_{λ} on G/B with fiber ℂ_{λ}. It is well-known that the correspondence λ → L_{λ} is an isomorphism of P and Pic(G/B). L_{λ} is ample if and only if L_{λ} is very ample if and only if λ ∈ P^{++}. By the Borel–Weil–Bott theorem [Bot], for strictly dominant λ the vector space of global sections H^{0}(G/B, L_{λ}) is isomorphic as a G-module to V_{λ}, the irreducible G-module with highest weight λ. The embedding G/B ⊂ ℙ(V_{λ^{∗}}) identifies G/B with the projectivization of the cone of highest weight vectors. The dual variety (G/B)^{∗} therefore lies in ℙ(V_{λ}) and parametrises global sections s ∈ H^{0}(G/B, L_{λ}) such that the scheme of zeros Z(s) is a singular divisor.

More generally, consider any flag variety of the form G/P, where P ⊂ G is an arbitrary parabolic subgroup. The subgroup P ⊂ G is called parabolic if G/P is a projective variety. P is parabolic if and only if it contains some Borel subgroup. Up to conjugacy, me may assume that P contains B. The combinatorial description is as follows. Let Π_{P} ⊂ Π be some subset of simple roots. Let Δ_{P}^{+} ⊂ Δ^{+} denote the positive roots that are linear combinations of the roots in Π_{P}. Then P is generated by B and by the root groups U_{α} for α ∈ Δ_{P}^{+}. We denote Π \ Π_{P} by Π_{G/P} and Δ^{+} \ Δ_{P}^{+} by Δ_{G/P}^{+} . A parabolic subgroup is maximal if and only if Π_{G/P} is a single simple root.

The fundamental weights ω_{i_{1}}, … ω_{i_{k}} dual to the simple roots in Π_{G/P} generate the sublattice P_{G/P} of P. We denote P^{+} ∩ P_{G/P} by P_{G/P}^{+} . The subset P_{G/P}^{++} ⊂ P_{G/P}^{+} consists of all weights λ = ∑ n_{k}ω_{i_{k}} such that all n_{k} > 0. Any weight λ ∈ P_{G/P} defines a character of P, and therefore a line bundle L_{λ} on G/P. Then the following is well-known.

– The correspondence λ → L_{λ} is an isomorphism of P_{G/P} and Pic(G/P). In particular, Pic(G/P) = ℤ if and only if P is maximal.

– L_{λ} is ample if and only if L_{λ} is very ample if and only if λ ∈ P_{G/P}^{++} .

– If λ ∈ P_{G/P}^{+} then the linear system corresponding to L_{λ} is base–point free. The corresponding map given by sections is a factorization G/P → G/Q, where Q is a parabolic subgroup such that Π_{Q} is a union of Π_{P} and all simple roots in Π_{G/P} orthogonal to λ.

– For strictly dominant λ ∈ P_{G/P}^{++} the vector space of global sections H^{0}(G/P, L_{λ}) is isomorphic as a G-module to V_{λ}, the irreducible G-module with highest weight λ.

The embedding G/P ⊂ ℙ(V_{λ^{∗}}) identifies G/P with the projectivization of the cone of highest weight vectors. The dual variety (G/P)^{∗} therefore lies in ℙ(V_{λ}) and parametrises global sections s ∈ H^{0}(G/P, L_{λ}) such that the scheme of zeros Z(s) is a singular divisor.

2.1.B Basic Examples

Example 2.1 (Projective spaces and Grassmanians) Let V = ℂ^{n} be a finite-dimensional vector space, G = SL(V). The projective space ℙ(V) is identified with a flag variety G/P, where P is the normalizer in G of a line in V. P is a maximal parabolic subgroup, therefore Pic(ℙ(V)) = ℤ. All line bundles on ℙ(V) have a form O(d) for d ∈ ℤ. The line bundle O(d) is ample if and only if it is very ample if and only if d > 0. In this case H^{0}(ℙ(V), O(d)) is canonically isomorphic to S^{d}V^{∗}. The embedding ℙ(V) ⊂ ℙ(S^{d}V) is called the Veronese embedding. It assigns to a line spanned by v ∈ V the line in S^{d}V spanned by v^{d}. The zero scheme Z(s) of the global section s ∈ S^{d}(V^{∗}) are just the hypersurface in ℙ(V) of degree d given by s = 0. Therefore, the dual variety in this case parametrises singular hypersurfaces. In particular, the corresponding discriminant is the classical discriminant of a homogeneous form of degree d in n variables.

The Grassmanian Gr(k, V) of k-dimensional vector subspaces of V is a flag variety G/P, where P is the normalizer in SL(V) of a k-dimensional subspace in V. The k-dimensional subspaces in V are in the bijective correspondence with (k − 1)-dimensional subspaces in ℙ(V). Having this in mind we shall sometimes write Gr(k − 1, ℙ(V)) instead of Gr(k, V). The Grassmanian Gr(k, V) carries a natural equivariant vector bundle S, called the tautological vector bundle. The fiber of S over a k-dimensional linear subspace U ⊂ V is U itself. Therefore, S is a subbundle of the trivial bundle Gr(k, V) × V with fiber V at each point.

P is a maximal parabolic subgroup corresponding to the fundamental weight ω_{k}, therefore Pic(Gr(k, V)) = ℤ. The ample generator of Pic(Gr(k, V)) is the line bundle Λ^{k}S^{∗}. The corresponding embedding is called the Plücker embedding, Gr(k, V) ⊂ ℙ(Λ^{k}V). Namely, if U ∈ Gr(k, V), u_{1}, …, u_{k} is any basis of U then U ↦ [u_{1}∧…∧u_{k}]. The discriminant of the Plücker embedding was studied by Lascoux [Las]. Clearly, all minimal flag varieties of SL(V) are isomorphic to Grassmanians.

The tangent space to Gr(k, V) at the point U is a vector space with an additional structure. Namely, it is easy to verify the natural isomorphism

T_{U} Gr(k, V) = Hom(U, V/U) = U^{∗} ⊗ V/U.

In particular, the tangent bundle T Gr(k, V) is canonically isomorphic to S^{∗} ⊗(V/S). Notice also that Gr(k, V) ≃ Gr(n−k, V^{∗}). Namely, any subspace U ∈ Gr(k, V) maps to its annihilator Ann U ∈ Gr(n − k, V^{∗}).

Example 2.2 (Spinor varieties) Let V = ℂ^{n} be a vector space equipped with a non-degenerate symmetric scalar product Q or a non-degenerate symplectic form ω (n should be even in this case). Let G = SO(V) or G = Sp(V), respectively. For any k ≤ n/2 let Gr_{Q}(k, V) (resp. Gr_{ω}(k, V)) denote the isotropic Grassmanian of k-dimensional isotropic subspaces. Recall that a subspace U ⊂ V is called isotropic if for any v_{1}, v_{2} ∈ U we have Q(v_{1}, v_{2}) = 0 (resp. ω(v_{1}, v_{2}) = 0). Then Gr_{Q}(k, V) and Gr_{ω}(k, V) are projective G-equivariant varieties. It easily follows from the Witt theorem that Gr_{Q}(k, V) and Gr_{ω}(k, V) are irreducible homogeneous varieties (hence flag varieties) in all cases except the symmetric case when n = 2k. In the latter case Gr_{Q}(k, ℂ^{2k}) consists of two homogeneous components Gr_{Q}^{+}(k, ℂ^{2k}) and Gr_{Q}^{−}(k, ℂ^{2k}). These two varieties are in fact isomorphic to each other (as algebraic varieties, not as algebraic varieties with the group G action). Indeed, let s be any orthogonal reflection. Then s(Gr_{Q}^{±}(k, ℂ^{2k})) = Gr_{Q}^{∓}(k, ℂ^{2k}). Moreover, Gr_{Q}^{±}(k, ℂ^{2k}) is isomorphic to Gr_{Q}(k − 1, ℂ^{2k−1}). To see this let H = ℂ^{2k−1} ⊂ ℂ^{2k} be any hyperplane such that the restriction of Q on H is non-degenerate. Then we have obvious maps

Gr_{Q}^{±}(k, ℂ^{2k}) → Gr_{Q}(k − 1, ℂ^{2k−1}), U ↦ U ∩ H.

It is easy to see that both maps are isomorphisms. The variety Gr_{Q}^{±}(k, ℂ^{2k}) is called the spinor variety, denoted by 𝕊_{k}.

The introduced varieties represent the complete list of minimal flag varieties of groups SO(V) and Sp(V). Namely, Gr_{ω}(k, V), k = 1, …, n/2, is the minimal flag variety of Sp(V) corresponding to the fundamental weight ω_{k} (for example, Gr_{ω}(1, V) = ℙ(V)). Gr_{Q}(k, V), 0 < k < n/2, is the minimal flag variety of SO(V) corresponding to the fundamental weight ω_{k} (for example, Gr_{Q}(1, V) is a quadric hypersurface in ℙ(V)). Finally, Gr_{Q}^{±}(k, ℂ^{2k}) is the minimal flag variety of SO(ℂ^{2k}) corresponding to the fundamental weights ω_{k−1} and ω_{k}.

In particular, the Picard group of all these varieties is isomorphic to ℤ. Let us describe the embedding corresponding to the ample generator of ℤ.

Consider the symplectic case G = Sp(V) first. Then we have the Plücker embedding Gr_{ω}(k, V) ⊂ ℙ(Λ^{k}V). However, Λ^{k}V is reducible as Sp(V)- module, it contains the submodule ω ∧ Λ^{k−2}V. The complement Λ_{0}^{k}V of this submodule is an irreducible module with highest weight ω_{k}. The Plücker embedding in fact gives the embedding of Gr_{ω}(k, V) in ℙ(Λ_{0}^{k}V) as the projectivization of the cone of highest weight vectors. This embedding corresponds to the ample generator of the Pickard group.

Now consider the orthogonal case G = SO(V). Then we have the Plücker embedding Gr_{Q}(k, V) ⊂ ℙ(Λ^{k}V). Suppose first that k < m if n = 2m + 1 or k < m − 1 if n = 2m. Then Λ^{k}V is irreducible as G-module and has the highest weight ω_{k}. Therefore, this embedding corresponds to the ample generator of the Pickard group.

Finally, the embedding of the spinor variety 𝕊_{m} corresponding to the ample generator of the Pickard group can be described as follows. Let W be the spinor representation of Spin(2m − 1) or the half-spinor representation of Spin(2m) (they are isomorphic with respect to the embedding Spin(2m−1) ⊂ Spin(2m)). Then 𝕊_{m} embeds into ℙ(W) as the projectivization of the highest weight vector orbit.

To give a more precise model, let us recall the definition of the half-spinor representation. Let V = ℂ^{2m} be an even-dimensional vector space with a nondegenerate symmetric scalar product Q. Let Cl(V) be the Clifford algebra of V. Recall that Cl(V) is in fact a superalgebra, Cl(V) = Cl^{0}(V) ⊕ Cl^{1}(V), where Cl^{0}(V) (resp. Cl^{1}(V)) is a linear span of elements of the form v_{1}·. . .·v_{r}, v_{i} ∈ V, r is even (resp. r is odd). Then we have

Spin(V) = {a ∈ Cl^{0}(V) | a = v_{1} · . . . · v_{r}, v_{i} ∈ V, Q(v_{i}, v_{i}) = 1}.

Spin(V) is a connected simply-connected algebraic group.

Given a ∈ Cl(V), a = v_{1} · . . . · v_{r}, v_{i} ∈ V, let \overline{a} = (−1)^{r}v_{r} · . . . · v_{1}. This is a well-defined involution of Cl(V). Using it, we may define an action R of Spin(V) on V by formula R(a)v = a · v · \overline{a}. Then R is a double covering Spin(V) → SO(V).

Let U ⊂ V be a maximal isotropic subspace, hence dim U = m. Take also any maximal isotropic subspace U′ such that U ⊕ U′ = V. For any v ∈ U we define an operator ρ(v) ∈ End(Λ^{∗}U) by the formula

ρ(v) · u_{1} ∧ … ∧ u_{r} = v ∧ u_{1} ∧ … u_{r}.

For any v ∈ U′ we define an operator ρ(v) ∈ End(Λ^{∗}U) by the formula

ρ(v) · u_{1} ∧ … ∧ u_{r} = ∑_{i=1}^{r}(−1)^{i−1}Q(v, u_{i}) ∧ u_{1} ∧ … u_{i−1} ∧ u_{i+1} ∧ … ∧ u_{r}.

These operators are well-defined and therefore by linearity we have a linear map ρ : V → End(Λ^{∗}U). It is easy to check that for any v ∈ V we have ρ(v)^{2} = Q(v, v) Id. Therefore we have a homomorphism of associative algebras Cl(V) → End(Λ^{∗}U), i.e., Λ^{∗}U is a Cl(V)-module, easily seen to be irreducible. Therefore, Λ^{∗}U is also a Spin(V)-module, now reducible. However, the even part Λ^{ev}U is an irreducible Spin(V)-module, called the halfspinor module S^{+}. Now let us describe the embedding of the spinor variety 𝕊_{m} ⊂ ℙ(S^{+}). This embedding will correspond to the ample generator of Pic(𝕊_{m}). By definition, 𝕊_{m} is an SO(V)-orbit of a fixed maximal isotropic subspace, say U′. We claim that the normalizer of U′ in Spin(V) (acting on V via the representation R) is equal to the normalizer of 1 ∈ Λ^{ev}U in Spin(V) (acting on Λ^{ev} via the representation ρ). This will show that 𝕊_{m} is naturally isomorphic to the projectivization of the orbit of 1 in the half-spinor representation. Consider the map

Ψ : V × Λ^{ev}U → Λ^{odd}U, (v, w) ↦ ρ(v)w.

Spin V acts on V via the representation R and on Λ^{ev} and Λ^{odd} via the representation ρ. We claim that Ψ is Spin(V)-equivariant. Indeed, for any g ∈ Spin(V) we have

Ψ(R(g)v, ρ(g)w) = ρ(R(g)v)(ρ(g)w) = ρ(gv\overline{g})(ρ(g)w)
= ρ(gv\overline{g}g)w = ρ(g)(ρ(v)w) = ρ(g)Ψ(v, w),

since g\overline{g} = 1. It is easy to check that Ψ(U′, 1) = 0 and if for any v ∈ U′ we have Ψ(v, w) = 0 then w = λ1 for some λ ∈ ℂ. Suppose now that g ∈ Spin(V) normalizes 1 ∈ Λ^{ev}U. Then for any v ∈ U′ we have

Ψ(R(g)v, ρ(g)1) = ρ(g)Ψ(v, 1) = 0,

therefore g normalizes U′. On the contrary, if g normalizes U′ then for any v ∈ U′

0 = Ψ(R(g)(R(g^{−1})v), ρ(g)1) = Ψ(v, ρ(g)1),

therefore g normalizes 1 ∈ Λ^{ev}U.

Example 2.3 (Severi varieties) Let 𝔻_{ℝ} denote one of four division algebras over ℝ: real numbers ℝ, complex numbers ℂ, quaternions ℍ, or octonions 𝕆. Let 𝔻 = 𝔻_{ℝ} ⊗_{ℝ} ℂ be its complexification, therefore 𝔻 is either ℂ, ℂ ⊕ ℂ, Mat_{2}(ℂ), or ℂa (the algebra of complex Cayley numbers). All these algebras have the standard involution. Let H_{3}(𝔻) denote the 3 × 3 Hermitian matrices over 𝔻. If x ∈ H_{3}(𝔻) then we may write

x = ( \begin{matrix}c_{1} & u_{1} & u_{2} \\ \overline{u_{1}} & c_{2} & u_{3} \\ \overline{u_{2}} & \overline{u_{3}} & c_{3} \end{matrix} ) , c_{i} ∈ ℂ, u_{i} ∈ 𝔻.

H_{3}(𝔻) is a Jordan algebra with respect to the operation x ∘ y = \frac{xy + yx}{2},

that is, for any a, b we have

a ∘ b = b ∘ a and ((a ∘ a) ∘ b) ∘ a = (a ∘ a) ∘ (b ∘ b),

see [J]. In case 𝔻 = ℂa this algebra is called the Albert algebra. The group of automorphisms G = Aut(H_{3}) is a semisimple algebraic group of type SO_{3}, SL_{3}, Sp_{6}, or F_{4}, respectively. The action of G on H_{3} has two irreducible components: scalar matrices and traceless matrices. It is possible to enlarge the group of automorphisms by considering the group of norm similarities. Namely, H_{3}(𝔻) has a cubic form det defined by the formula

det(x) = \frac{1}{6} ((Tr x)^{3} + 2 Tr(x^{3}) − 3(Tr x) Tr(x^{2})) .

The group of norm similarities G̃ is defined as a subgroup in SL(H_{3}) preserving det. G̃ is a semisimple algebraic group of type SL_{3}, SL_{3} × SL_{3}, SL_{6}, or E_{6}, respectively. The representation of G̃ on H_{3} is irreducible. The projectivization of the highest weight vector orbit is called the Severi variety corresponding to 𝔻. One may think of Severi varieties as of complexifications of projective planes over 𝔻_{ℝ}.

The Severi variety corresponding to ℂa is the minimal flag variety of E_{6} (corresponding to the fundamental weight ω_{1}) embedded into ℙ^{26} by the ample generator of its Picard group. The corresponding elements of the Albert algebra are known as elements of rank 1, see [J]. Other Severi varieties have the following description.

If 𝔻_{ℝ} = ℝ then H_{3} is equal to the Jordan algebra of symmetric 3 × 3 complex matrices and det is the standard determinant. G̃ = SL_{3} acts on symmetric matrices as on quadratic forms, the Severi variety is the projectivization of the cone of rank 1 symmetric matrices. Therefore, this Severi variety is isomorphic to ℙ^{2} in the second Veronese embedding.

If 𝔻_{ℝ} = ℂ then H_{3} is equal to the Jordan algebra of 3×3 complex matrices and det is the standard determinant. G̃ = SL_{3} × SL_{3} acts on matrices by left and right multiplication, the Severi variety is the projectivization of the cone of rank 1 matrices. Therefore, this Severi variety is isomorphic to ℙ^{2} × ℙ^{2} in the Segre embedding.

If 𝔻_{ℝ} = ℍ then H_{3} is equal to the Jordan algebra Λ^{2}ℂ^{6} with the multiplication given by α_{1}∘α_{2} = α_{1}∧α_{2}, where we identify Λ^{2}ℂ^{6} and Λ^{4}ℂ^{6} by means of a standard symplectic form ω (it is easy to see that Λ^{4}ℂ^{6} = ω∧Λ^{2}ℂ^{6}). The function det is defined as det(α) = α ∧ α ∧ α (in the matrix form det is equal to the Pfaffian of the skew-symmetric matrix corresponding to α). G̃ = SL_{6} acts on Λ^{2}ℂ^{6} traditionally. The Severi variety is the projectivization of the variety of decomposable bivectors of the form u∧v, u, v ∈ ℂ^{6}. Therefore, this Severi variety is isomorphic to Gr(2, ℂ^{6}) in the Plücker embedding.

For 𝔻_{ℝ} equal to ℝ, ℂ, or ℍ it is possible to define higher dimensional Severi varieties in the same way (isomorphic to ℙ^{n} in the second Veronese embedding, ℙ^{n} × ℙ^{n} in the Segre embedding, or Gr(2, ℂ^{2n+2}) in the Plücker embedding). However, for 𝔻_{ℝ} = 𝕆 only the case of 3 × 3 matrices makes sense.

Example 2.4 (Adjoint varieties) For the adjoint representation of SL_{n} = SL(V) there is a natural notion of the discriminant defined as follows. For any operator A ∈ 𝔰𝔩(V) let P_{A} = det(t Id −A) be the characteristic polynomial. Then its discriminant, D(A) = D(P_{A}) is a homogeneous form on 𝔰𝔩(V) of degree n^{2} − n. Clearly D(A) ≠ 0 if and only if all eigenvalues of A are distinct, that is, if A is a regular semisimple operator. This notion can be carried over any simple Lie algebra. Before doing that, notice that D(A) can be also defined as follows. Consider the characteristic polynomial

Q_{A} = det(t Id − ad(A)) = ∑_{i=0}^{n^{2}−1} t^{i}D_{i}(A)

of the adjoint operator ad(A). If λ_{1}, …, λ_{r} are eigenvalues of A (counted with multiplicities) then the set of eigenvalues of ad(A) consists of n − 1 zeros and λ_{i} − λ_{j}, i, j = 1, …, n, i ≠ j. Therefore, D_{0}(A) = … = D_{n−2}(A) = 0 and D_{n−1}(A) coincides with D(A) up to a non-zero scalar.

Suppose now that 𝔤 is a simple Lie algebra of rank r. Let dim 𝔤 = n. For any x ∈ 𝔤 let

Q_{x} = det(t Id − ad(x)) = ∑_{i=0}^{n} t^{i}D_{i}(x)

be the characteristic polynomial of the adjoint operator. Then D(x) = D_{r}(x) is called the discriminant of x. Clearly, D is a homogeneous Ad-invariant polynomial on 𝔤 of degree n − r. Since the dimension of the centralizer 𝔤_{x} of any element x ∈ 𝔤 is greater or equal to r, it follows that D_{0} = … = D_{r−1} = 0, therefore D(x) = 0 if and only if ad(x) has the eigenvalue 0 with multiplicity > r. We claim that actually D(x) ≠ 0 if and only if x is regular semisimple (recall that x is called regular if dim 𝔤_{x} = r). Indeed, if x is semisimple then ad(x) is a semisimple operator, therefore D(x) = 0 if and only if the dimension of the centralizer 𝔤_{x} is greater than r, i.e. x is not regular. If x is not semisimple then we take the Jordan decomposition x = x_{s} + x_{n}, where x_{s} is semisimple, x_{n} is nilpotent, and [x_{s}, x_{n}] = 0. Then x_{s} is automatically not regular, therefore since Q_{x} = Q_{x_{s}} we have D(x) = D(x_{s}) = 0 (of course, this last equality also follows from the Adinvariance of D and the fact that the Ad-orbit of x contains the Ad-orbit of x_{s} in its closure).

To study D(x) further, we can use the Chevalley restriction theorem (see [PV]) ℂ[𝔤]^{G} = ℂ[𝔱]^{W}, where 𝔱 ⊂ 𝔤 is any Cartan subalgebra and W is the Weil group. Let Δ ⊂ 𝔱^{∗} be the root system, #Δ = n − r. Then, clearly, for any x ∈ 𝔱 D(x) = 0 if and only if x is not regular if and only if α(x) = 0 for some α ∈ Δ. Since deg D = n − r and D|_{𝔱} is W -invariant, it easily follows that

D|_{𝔱} = ∏_{α∈Δ} α.

The Weyl group acts transitively on the set of roots of the same length. Therefore, D|_{𝔱}, and hence D, is irreducible if and only if all roots in Δ have the same length, i.e. Δ is of type A, D, or E. If Δ is of type B, C, F, or G, we have Δ = Δ_{s} ∪ Δ_{l}, where Δ_{s} is the set of short roots and Δ_{l} is the set of long roots. Then we have D = D_{l}D_{s}, where D_{l} and D_{s} are irreducible polynomials and

D_{l}|_{𝔱} = ∏_{α∈Δ_{l}} α, D_{s}|_{𝔱} = ∏_{α∈Δ_{s}} α.

In the A − D − E case we also set D_{l} = D to simplify notations.

We are going to show that D_{l} is also the discriminant in our regular sense. The adjoint representation Ad : G → GL(𝔤) is irreducible. In A − D − E case let O = O_{l} be the Ad-orbit of any root vector, In B − C − F − G case let O_{l} ⊂ 𝔤 (resp. O_{s} ⊂ 𝔤) be the Ad-orbit of any long root vector (resp. any short root vector). Then O_{l} is the orbit of the highest weight vector. Its projectivization is called the adjoint variety. Both orbits O_{l} and O_{s} are conical. Let X_{l} = ℙ(O_{l}), X_{s} = \overline{ℙ(O_{s})}.

Theorem 2.5 D_{l} is the discriminant of X_{l}. D_{s} is the discriminant of X_{s}.

Proof. We identify 𝔤 and 𝔤^{∗} via the Killing form. Let α ∈ Δ be any root, e_{α} ∈ 𝔤 the corresponding root vector. Since [𝔤, e_{α}]^{⊥} = 𝔤_{e_{α}}, we have

(\overline{ℙ(Ad(G) · e_{α})})^{∗} = \overline{ℙ(Ad(G) · 𝔤_{e_{α}})}.

Let

𝔱_{α} = {x ∈ 𝔱 | α(x) = 0}, 𝔱̂_{α} = 𝔱_{α} + e_{α}.

Then, clearly, 𝔱̂_{α} ∈ 𝔤_{e_{α}}. Moreover, if α is long then D_{l}|𝔱̂_{α} = D_{l}|_{𝔱_{α}} = 0. If α is short then D_{s}|𝔱̂_{α} = D_{s}|_{𝔱_{α}} = 0. Therefore, in order to prove Theorem it suffices to check that

dim Ad(G) · 𝔱̂_{α} = n − 1.  (2.1)

Clearly, for generic x ∈ 𝔱̂_{α} we have

dim Ad(G) · 𝔱̂_{α} = dim G + dim 𝔱̂_{α} − dim Tran(x, 𝔱̂_{α}),  (2.2)

where

Tran(x, 𝔱̂_{α}) = {g ∈ G | Ad(g)x ∈ 𝔱̂_{α}}.

Let x = y + e_{α} ∈ 𝔱̂_{α}, where y ∈ 𝔱_{α} is such that for any β ∈ Δ \ {±α} we have β(y) ≠ 0. Suppose that Ad(g)x ∈ 𝔱̂_{α}. Since [e_{α}, 𝔱_{α}] = 0, it easily follows that Ad(g)y ∈ 𝔱_{α} and Ad(g)e_{α} = e_{α}. Under our assumptions on y this means that

Tran(x, 𝔱̂_{α}) = (N_{G}(𝔱_{α}))_{e_{α}},

where N_{G}(𝔱_{α}) is the normalizer of t_{α} in G. Therefore, in order to prove (2.1) using (2.2), it suffices to check that

dim(𝔤_{𝔱_{α}})_{e_{α}} = r,  (2.3)

where 𝔤_{𝔱_{α}} is the centralizer of t_{α} in 𝔤. Now, 𝔤_{𝔱_{α}} is a Levi subalgebra equal to 𝔱 + ℂe_{α} + ℂe_{−α}. Therefore, (𝔤_{𝔱_{α}})_{e_{α}} = 𝔱_{α} + ℂe_{α} and (2.3) follows. ⊓ ⊔

2.2 The Pyasetskii Pairing

2.2.A Actions With Finitely Many Orbits

In [Py] Pyasetskii has shown that if a connected algebraic group acts linearly on a vector space with a finite number of orbits then the dual representation has the same property and, moreover, the number of orbits is the same. In this section we deduce this result from the projective duality and give some examples and applications.

Theorem 2.6 Suppose that a connected algebraic group G acts on a projective space ℙ^{n} with a finite number of orbits. Then the dual action G : ℙ^{n}^{∗}

has the same number of orbits. Let ℙ^{n} = ⊔_{i=1}^{N} O_{i} and ℙ^{n}^{∗} = ⊔_{i=1}^{N} O′_{i} be the orbit

decompositions. Let O_{0} = O′_{0} = ∅. Then the bijection is defined as follows: O_{i} corresponds to O′_{j} if and only if \overline{O_{i}} is projectively dual to \overline{O′_{j}}.

Proof. Indeed, take any G-orbit O ⊂ ℙ^{n}^{∗}. If \overline{O} ≠ ℙ^{n}^{∗} then the projectively dual variety \overline{O}^{∗} is the closure of some orbit O′ ⊂ ℙ^{n}, since it is G-invariant. Therefore, by the Reflexivity Theorem we have \overline{O} = \overline{O′}^{∗}. ⊓ ⊔

Remark 2.7 If the number of orbits for the action G : ℙ^{n} is infinite then, in general, there is no natural bijection between orbits in ℙ^{n} and ℙ^{n}^{∗}. However, it can be shown (see e.g. [Py]) that the modalities of these actions are equal (modality is equal to the maximal number of parameters that the family of orbits can depend on). This also easily follows from the arguments involving projective duality.

The linear action of an algebraic group on a vector space V is called conical if any G-orbit O is conical, i.e. O is preserved by homotheties. The action is conical if and only if for any v ∈ V, v is contained in the tangent space 𝔤 · v to the orbit Gv. For example, if there are only finitely many orbits then the action is conical. If the action is conical then its non-zero orbits are in a 1-1 correspondence with orbits of the projectivization G : ℙ(V).

Proposition 2.8 If the linear action G : V is conical then the dual action G : V^{∗} is conical as well.

Proof. We can embed V = ℂ^{n} and V^{∗} = (ℂ^{n})^{∗} in projective spaces ℙ^{n} and ℙ^{n}^{∗} as the affine charts. These embeddings can be done equivariantly. Points of V^{∗} correspond to hyperplanes in ℙ^{n} not passing through the origin 0 ∈ V ⊂ ℙ^{n}. Suppose that O ⊂ V^{∗} is a non-conical orbit. Then the dual variety \overline{O}^{∗} ⊂ ℙ^{n} intersects with ℂ^{n} non-trivially. Therefore, \overline{O}^{∗} ⊂ ℙ^{n} is the closure of a conical variety in ℂ^{n}. Therefore, its dual variety \overline{O}^{∗}^{∗} ⊂ ℙ^{n}^{∗} does not intersect (ℂ^{n})^{∗}. But this contradicts the Reflexivity Theorem. ⊓ ⊔

Now the Pyasetskii Theorem is an easy corollary of Theorem 2.6 and Proposition 2.8.

Corollary 2.9 ([Py]) Suppose that a connected algebraic group G acts linearly on a vector space V with a finite number of orbits. Then the dual action

G : V^{∗} has the same number of orbits. Let V = ⊔_{i=1}^{N} O_{i} and V^{∗} = ⊔_{i=1}^{N} O′_{i} be the orbit decompositions. Then the bijection is defined as follows: O_{i} corresponds to O′_{j} if and only if ℙ(\overline{O_{i}}) is projectively dual to ℙ(\overline{O′_{j}}).

Recall that the Reflexivity Theorem is equivalent to the formula (1.2) that Lag(O_{i}) = Lag(O′_{j}), where for any conical variety Y ⊂ V we denote by Lag(Y) the closure of the conormal bundle N_{Y_{sm}}^{∗}V in the cotangent bundle T^{∗}V = V ⊕V^{∗}. The varieties Lag(O_{i}), i = 1, …, N have a nice interpretation. Namely, the action of 𝔤 (the Lie algebra of G) on V is given by the element of

Hom(𝔤, End(V)) = 𝔤^{∗} ⊗ V^{∗} ⊗ V = Hom(V^{∗} ⊗ V, 𝔤^{∗}).

The corresponding bilinear map μ : V ⊕ V^{∗} → 𝔤^{∗} is called the moment map. Let L = μ^{−1}(0) be its zero fiber. Then it is easy to see that L = ∪_{i} Lag(O_{i}) is the decomposition of L into irreducible components.

The large class of linear actions of reductive groups with finitely many orbits is provided by graded semisimple Lie algebras. Suppose that 𝔤 = ⊕_{k∈ℤ}𝔤_{k} is a graded semi-simple Lie algebra of the connected semi-simple group G. Then there exists a unique semisimple element ξ ∈ 𝔤_{0} such that

g_{k} = {x ∈ 𝔤 | [ξ, x] = kx}.

The connected component H of the centralizer G_{ξ} ⊂ G is a reductive subgroup of G called the Levi subgroup (because it is the Levi part of the parabolic subgroup). 𝔤_{0} is the Lie algebra of H. H acts on each graded component 𝔤_{k}.

Theorem 2.10 ([Ri, Vi]) H acts on 𝔤_{k} with finitely many orbits.

Proof. Clearly all elements of 𝔤_{k} are nilpotent in 𝔤. Since G has finitely many nilpotent orbits, it is sufficient to prove that for any nilpotent G-orbit O the intersection O ∩ 𝔤_{k} consists of finitely many H-orbits. By the general Richardson Lemma (see [Ri, PV]), it is enough to check that for any x ∈ O∩𝔤_{k} the tangent space [𝔤_{0}, x] to the H-orbit Ad(H)x is equal to the intersection of 𝔤_{k} with the tangent space [𝔤, x] to the G-orbit Ad(G)x. But this is clear:

[𝔤, x] ∩ 𝔤_{k} = [⊕_{i}𝔤_{i}, x] ∩ 𝔤_{k} = ⊕_{i}[𝔤_{i}, x] ∩ 𝔤_{k} = [𝔤_{0}, x].

Theorem is proved. ⊓ ⊔

Notice that the dual H-module (𝔤_{k})^{∗} is isomorphic to 𝔤_{−k} under the Killing form and the moment map 𝔤_{k} × 𝔤_{−k} → 𝔤_{0} is given by the Lie bracket.

2.2.B The Multisegment Duality.

Apart from actions associated with ℤ-graded semisimple Lie algebras, another class of actions with finitely many orbits is provided by the theory of representations of quivers (see [Ga]). These two classes overlap: the representations of quivers of type A give the same class of actions as the standard gradings of of SL(V). The Pyasetskii pairing in this case was studied in a series of papers under the name of the multisegment duality, or the Zelevinsky involution. Here we give an overview of some of these results.

We fix a positive integer r and consider the set S = S_{r} of pairs of integers (i, j) such that 1 ≤ i ≤ j ≤ r. Let ℤ_{+}^{S} denote the semigroup of families m = (m_{ij})_{(i,j)∈S} of non-negative integers indexed by S. The set S and the semigroup ℤ_{+}^{S} have several interpretations. First, S_{r} is naturally identified with the set of positive roots of type A_{r}. Namely, if α_{1}, …, α_{r} denote simple roots then each (i, j) ∈ S corresponds to a positive root α_{i} + α_{i+1} + … + α_{j}. Another useful interpretation is to regard a pair (i, j) ∈ S as a segment [i, j] = {i, i + 1, …, j} in ℤ. A family m = m_{ij} ∈ ℤ_{+}^{S} can be regarded as a collection of segments, containing m_{ij} copies of each [i, j]. Thus, elements of ℤ_{+}^{S} can be called multisegments. The weight |m| of a multisegment m is defined as a sequence γ = {d_{1}, …, d_{r}) ∈ ℤ_{+}^{r} given by

d_{i} = ∑_{i∈[k,l]} m_{kl} for i = 1, …, r.

In other words, |m| records how many segments of m contain any given number i ∈ [1, r]. For any γ ∈ ℤ_{+}^{r} we set ℤ_{+}^{S}(γ) = {m ∈ ℤ_{+}^{S} | |m| = γ}. In the language of positive roots, ℤ_{+}^{r} is the semigroup generated by simple roots. Under this identification, the elements m ∈ ℤ_{S}^{+}(γ) become the partitions of γ into a sum of positive roots. The cardinality of ℤ_{S}^{+}(γ) is known as a Kostant partition function (of γ).

Another important interpretation of ℤ_{S}^{+}(γ) is that it parametrizes isomorphism classes of representations of quivers of type A. Let A_{r} be the quiver equal to the Dynkin diagram of type A_{r}, where all edges are oriented from the left to the right. Let A_{r}^{∗} be a dual quiver with all orientations reversed. The representation of A_{r} with the dimension vector γ = {d_{1}, …, d_{r}) ∈ ℤ_{+}^{r} is the collection of vector spaces ℂ^{d_{1}}, …, ℂ^{d_{r}} and linear maps

φ_{1} : ℂ^{d_{1}} → ℂ^{d_{2}}, …, φ_{r−1} : ℂ^{d_{r−1}} → ℂ^{d_{r}}.

The representation of A_{r}^{∗} with the dimension vector γ = {d_{1}, …, d_{r}) ∈ ℤ_{+}^{r} is the collection of vector spaces ℂ^{d_{1}}, …, ℂ^{d_{r}} and linear maps

φ_{1} : ℂ^{d_{2}} → ℂ^{d_{1}}, …, φ_{r−1} : ℂ^{d_{r}} → ℂ^{d_{r−1}}.

Therefore, representations of A_{r} with dimension vector γ are parametrized by points of a vector space V (γ) = ⨁_{i=1}^{r−1} Hom(ℂ^{d_{i}}, ℂ^{d_{i+1}}). Representations of A_{r}^{∗} with dimension vector γ are parametrized by points of a vector space V (γ)^{∗} = ⨁_{i=1}^{r−1} Hom(ℂ^{d_{i+1}}, ℂ^{d_{i}}). Notice that vector spaces V (γ) and V (γ)^{∗} are naturally dual to each other. Moreover, V (γ) and V (γ)^{∗} are dual modules of the group G(γ) = GL_{d_{1}} × … × GL_{d_{r}} with respect to the natural action. These actions could also be described in terms of graded Lie algebras. Namely, consider the Lie algebra 𝔤 = 𝔤𝔩_{d_{1}+...+d_{r}}. Elements of 𝔤 can be represented as block matrices with diagonal blocks of shapes d_{1} × d_{1}, …, d_{r} × d_{r}. This determines ℤ-grading: block diagonal matrices have grade 0, etc. Then 𝔤_{0} is a Lie algebra of G(γ) and the action of G(γ) on V (γ) and V (γ)^{∗} is equivalent to the action of G(γ) on 𝔤_{−1} and 𝔤_{1}.

The orbits of G(γ) on V (γ) (resp. V (γ)^{∗}) correspond to isoclasses of representations of A_{r} (resp. A_{r}^{∗}) with dimension vector γ. These orbits are parametrized by elements of ℤ_{S}^{+}(γ). Elements of S parametrize indecomposable A_{r}-modules (or A_{r}^{∗}-modules). Namely, each (i, j) ∈ S corresponds to an indecomposable module R_{ij} with dimension vector

|(i, j)| = (0^{i−1}, 1^{j−i+1}, 0^{r−j})

and all maps are isomorphisms, if it is possible, or zero maps otherwise. Then any family (m_{ij}) ∈ ℤ_{S}^{+} corresponds to an A_{r} (or A_{r}^{∗}) module ⊕_{S}R_{ij}^{m_{ij}}.

By Pyasetskii Theorem 2.9, there is a natural bijection of G(γ)-orbits in V (γ) and V (γ)^{∗}. Therefore, there exists a natural involution ζ of ℤ_{+}^{S}(γ), which can be extended to a weight-preserving involution of ℤ_{+}^{S} called the multisegment duality.

An explicit description of ζ was found in [KZ] using the Poljak Theorem [Po] from the combinatorial theory of networks. To formulate it, we need the following definition. For any multisegment m ∈ ℤ_{+}^{S} the ranks r_{ij}(m) are given by

r_{ij}(m) = ∑_{[i,j]⊂[k,l]} m_{kl}.

It is easy to see that the multisegment m can be recovered from its ranks by formula

m_{ij} = r_{ij}(m) − r_{i−1,j}(m) + r_{i−1,j+1}(m).

If the multisegment corresponds to the representation of A_{r} given by

(φ_{1}, …, φ_{r−1}) ∈ V (γ)

then r_{ij} is equal to the rank of the map φ_{j−1} ∘ … ∘ φ_{i+1} ∘ φ_{i}. In particular, r_{ii} = d_{i}.

For any (i, j) ∈ S let T_{ij} denote the set of all maps ν : [1, i]×[j, r] → [i, j] such that ν(k, l) ≤ ν(k′, l′) whenever k ≤ k′, l ≤ l′ (in other words, ν is a morphism of partially ordered sets, where [1, i] × [j, r] is supplied with the product order).

Theorem 2.11 ([KZ]) For every m = (m_{ij}) ∈ ℤ_{+}^{S} we have

r_{ij}(ζ(m)) = min_{ν∈T_{ij}} ∑_{(k,l)∈[1,i]×[j,r]} m_{ν(k,l)+k−i,ν(k,l)+l−j}.

The inductive description of ζ was given in [MW]. Let m = (m_{ij}) be a multisegment of weight γ = (d_{1}, …, d_{r}). We set i_{1} = min{i | d_{i} ≠ 0} and define the sequence of indices j_{1}, …, j_{p} as follows:

j_{1} = min{j | m_{i_{1}j} ≠ 0}, …, j_{t+1} = min{j | j > j_{t}, m_{i_{1}+t,j} ≠ 0},

where t = 1, …, p − 1. The sequence terminates when j_{p+1} does not exist. Let i_{t} = i_{1} + t − 1 for t = 1, …, p + 1. We associate to m the multisegment m′ given by

m′ = m − (i_{1}, j_{1}) − (i_{2}, j_{2}) − … − (i_{p}, j_{p}) + (i_{2}, j_{1}) + (i_{3}, j_{2}) + … + (i_{p+1}, j_{p}),

where we use the convention that (i, j) = 0 unless 1 ≤ i ≤ j ≤ r.

Theorem 2.12 ([MW]) If the multisegment m′ is associated to m then

ζ(m) = ζ(m′) + (i_{1}, i_{p}).

The involution ζ can also be described in terms of irreducible finitedimensional representations of affine Hecke algebras and in terms of canonical bases for quantum groups, see [KZ].

2.3 Parabolic Subgroups With Abelian Unipotent Radical

Let L be a simple algebraic group and P ⊂ L a parabolic subgroup with abelian unipotent radical. In this case 𝔩 = Lie L admits a ℤ-grading with only three non-zero parts:

𝔩 = 𝔩_{−1} ⊕ 𝔩_{0} ⊕ 𝔩_{1}.

Such a grading is said to be short. Here 𝔩_{0} ⊕ 𝔩_{1} = Lie P and exp(𝔩_{1}) is the abelian unipotent radical of P.

There exists a unique semisimple element ξ ∈ 𝔩_{0} such that

l_{k} = {x ∈ 𝔩 | [ξ, x] = kx}.

We denote by G the connected component of the centralizer L_{ξ} ⊂ L. Then 𝔩_{0} = 𝔤 = Lie G and by Theorem 2.10 G acts on 𝔩_{±1} with a finite number of orbits. In this section we describe explicitly these G-orbits and the Pyasetskii pairing. These results are well-known, we follow mainly [Pan] and [MRS].

We denote by T the maximal torus in G (and hence in L), 𝔱 = Lie T, Δ is the root system of (𝔩, 𝔱),

Δ = Δ_{−1} ∪ Δ_{0} ∪ Δ_{1}

is the partition corresponding to the short grading. We fix a Borel subgroup B ⊂ G containing T. This choice determines a set of positive roots

Δ_{0}^{+} ⊂ Δ_{0} and Δ^{+} = Δ_{0}^{+} ∪ Δ^{1} ⊂ Δ.

We denote by γ the highest root in Δ^{+}, in fact in Δ_{1}. Π is the set of simple roots in Δ^{+}, then Π_{0} = Π ∩ Δ_{0} is the set of simple roots in Δ_{0}^{+}. Let W (resp. W_{0}) be the Weil group of 𝔩 (resp. of 𝔤) with respect to 𝔱. For α ∈

Δ, we let w_{α} denote the corresponding reflection in W, α^{∨} = \frac{2α}{(α, α)} the

corresponding coroot, and e_{α} a non-zero root vector.

Proposition 2.13 The representation of 𝔤 on 𝔩_{1} is faithful and irreducible.

Proof. Indeed, 𝔨 = {x ∈ 𝔤 | ad(x)𝔩_{1} = 0} is an ideal in 𝔤. Since 𝔩_{1} and 𝔩_{−1} are dual 𝔤-modules, [𝔨, 𝔩_{−1}] = 0. It follows that 𝔨 is an ideal in 𝔩. Therefore, 𝔨 = 0 since 𝔩 is simple.

Suppose now that 𝔩_{1} is not irreducible, 𝔩_{1} = 𝔩_{1}^{1} ⊕𝔩_{1}^{2} is the 𝔤-module decomposition. Since 𝔩_{−1} is dual to 𝔩_{1}, we have 𝔩_{−1} = 𝔩_{−1}^{1} ⊕ 𝔩_{−1}^{2}, where 𝔩_{−1}^{i} = (𝔩_{1}^{j})^{⊥} for i ≠ j. Then [𝔩_{−1}^{i}, 𝔩_{1}^{j}] = 0 for i ≠ j. Indeed, for any x ∈ 𝔩_{−1}^{i}, y ∈ 𝔩_{0}, z ∈ 𝔩_{1}^{j} we have (y, [x, z]) = −([x, y], z) = 0, therefore [x, z] = 0. It follows that 𝔩_{−1}^{i} ⊕ [𝔩_{−1}^{i}, 𝔩_{1}^{i}] ⊕ 𝔩_{1}^{i} is an ideal in l. But 𝔩 is simple. Contradiction. ⊓ ⊔

Therefore, the center of 𝔤 is one-dimensional, spanned by ξ, P is a maximal parabolic subgroup, and #Π_{0} = #Π − 1. Thus, there is a unique simple root in Δ_{1}. Call it β.

Proposition 2.14

(a) β is long.

(b) The β-height of any root α_{0} ∈ Δ_{1}, i.e. the coefficient n_{β} in the sum α_{0} = n_{β}β + ∑_{α∈Π_{0}} n_{α}α, is equal to 1.

(c) For any α, α′ ∈ Δ_{1}, (α, α′) ≥ 0.

(d) W_{0} acts transitively on long roots in Δ_{1}.

Proof. (a) Indeed, since 𝔩_{1} is an irreducible 𝔤-module, β is the unique lowest weight of the 𝔤-module 𝔩_{1}. The longest element in W_{0} takes β to the highest weight of 𝔩_{1}, i.e. to γ. Hence β is long.

(b) Since 𝔩_{1} = U𝔤 · β, the β-height of any root in Δ_{1} is equal to 1.

(c) Suppose that α, α′ ∈ Δ_{1}. Since [e_{α}, e_{α′}] = 0, it follows that α + α′ is not a root, therefore, (α, α′) ≥ 0.

(d) Suppose that α_{0} ∈ Δ_{1} is a long root, α_{0} = β + ∑_{α∈Π_{0}} n_{α}α. If we have (α_{0}, α) < 0 for some α ∈ Π_{0} then w_{α}(α_{0}) > α_{0} and we may finish by induction. Suppose that (α_{0}, α) ≥ 0 for all α ∈ Π_{0}. Since (α_{0}, β) ≥ 0 by (c), α_{0} is a dominant weight for 𝔩. Since α_{0} is long, it follows that α_{0} = δ. Therefore, all long roots in Δ_{1} are W_{0}-conjugate to δ. ⊓ ⊔

On the contrary, if 𝔩 is an arbitrary simple Lie algebra, Δ ⊃ Π is the root system and the simple roots, β ∈ Π, and the β-height of the highest root is equal to 1, then, clearly, the maximal parabolic subgroup corresponding to β has an abelian unipotent radical. Using this observation, it is easy to find all parabolic subgroups with abelian unipotent radical in simple algebraic groups. If β = α_{k} in the Bourbaki numbering of simple roots then we shall denote the corresponding parabolic subgroup by P_{k}.

– A_{r}: any maximal parabolic subgroup P has an abelian unipotent radical.

– B_{r}: the only possibility is P_{1}.

– C_{r}: the only parabolic subgroup with abelian unipotent radical is P_{r}.

– D_{r}: there are three possibilities: P_{1}, P_{r−1}, and P_{r}

– Finally, there are two exceptional cases: P_{1} (or P_{6}) of E_{6} and P_{7} of E_{7}.

Most results about shortly graded simple Lie algebras can be proved by induction using the following procedure. Let

Δ′ = {α ∈ Δ | (α, β) = 0}.

Then Δ′ = Δ′_{−1} ∪ Δ′_{0} ∪ Δ′_{1} is the root system of the graded semisimple subalgebra 𝔩′ ⊂ 𝔩.

Proposition 2.15 Suppose that Δ′_{1} ≠ ∅. Then

(a) δ ∈ Δ′_{1}.

(b) 𝔩′_{0} = 𝔩′′_{0} ⊕ 𝔩′′′_{0}, where 𝔩′_{−1} ⊕ 𝔩′′_{0} ⊕ 𝔩′_{1} is a shortly graded simple Lie algebra.

(c) Δ′_{1} contains a unique minimal root, say β′. β′ is long.

Proof. Indeed, suppose that (β, α_{0}) = 0 for some α_{0} ∈ Δ_{1}. Then

(β, δ) = (β, α_{0}) + (β, ∑_{α∈Π_{0}} n_{α}α) ,

where all n_{α} ≥ 0. Since (β, α) ≤ 0 for any α ∈ Π_{0}, we get (β, δ) ≤ 0. Since (β, δ) ≥ 0 by Proposition 2.14, we see that δ ∈ Δ′_{1}. It follows that δ is the unique highest weight in 𝔩′_{0}-module 𝔩′_{1}. Therefore, 𝔩′_{1} is an irreducible 𝔩′_{0}- module and we obtain (b). Now we can apply Proposition 2.14 to the shortly graded simple Lie algebra 𝔩′_{−1} ⊕ 𝔩′′_{0} ⊕ 𝔩′_{1} and get (c). ⊓ ⊔

It is possible to define two canonical strings of pairwise orthogonal long roots in Δ_{1}. The construction is originally due to Harish Chandra. The lower canonical string β_{1}, …, β_{r} is the cascade up from β_{1} = β within Δ_{1}: at each stage β_{i+1} is the minimal root in

Δ_{1}^{(i)} = {α ∈ Δ_{1} | (α, β_{1}) = … = (α, β_{i}) = 0}.

The process terminates when Δ_{1}^{i} is empty. The previous proposition shows that the lower canonical string β_{1}, …, β_{r} is well-defined, i.e. on each step Δ_{1}^{i} contains a unique minimal element. Clearly, all the roots β_{i} are long.

The upper canonical string γ_{1}, …, γ_{r′} is the cascade down from γ_{1} = γ within Δ_{1}. At each stage, γ_{i+1} is the unique maximal element in

(i)Δ_{1} = {α ∈ Δ_{1} | (α, γ_{1}) = … = (α, γ_{i}) = 0}.

The process terminates when ^{(i)}Δ_{1} becomes empty. Obviously, the longest element in W_{0} takes the lower canonical string to the upper canonical string. Hence, the upper canonical string is well-defined and both strings have the same cardinality. In fact, we have the following proposition

Proposition 2.16 ([RRS]) Suppose that β′_{1}, …, β′_{r′} is a string of pairwise orthogonal long roots in Δ_{1}. Then r′ ≤ r and there exists w ∈ W_{0} such that β′_{i} = w(β_{i}) for i ≤ r′.

Proof. By Proposition 2.14 (d) we may suppose that β′_{1} = β_{1}. Now the proof goes by induction: both β′_{2}, …, β′_{r′} and β_{2}, …, β_{r} belong to Δ′_{1} and, hence, there exists w ∈ W′_{0} (the Weil group of 𝔩′_{0}) such that β′_{i} = w(β_{i}) for i ≤ r′. It remains to notice that wβ_{1} = β_{1}. ⊓ ⊔

Now we can describe the orbit decomposition and the Pyasetskii pairing.

Theorem 2.17 Let α_{1}, …, α_{r} ∈ Δ_{1} be any maximal sequence of pairwise orthogonal long roots. Set e_{k} = e_{α_{1}} + … + e_{α_{k}} for k = 0, …, r (so e_{0} = 0). Denote the G-orbit of e_{k} by O_{k} ⊂ 𝔩_{1}. Set f_{k} = e_{−α_{1}} + … + e_{−α_{k}} for k = 0, …, r (so f_{0} = 0). Denote the G-orbit of f_{k} by O′_{k} ⊂ 𝔩_{−1}. Then

(a) 𝔩_{1} = ⊔_{i=0}^{r} O_{i}, 𝔩_{−1} = ⊔_{i=0}^{r} O′_{i}.

(b) O_{i} ⊂ \overline{O_{j}} if and only if i ≤ j if and only if O′_{i} ⊂ \overline{O′_{j}}.

(c) O_{k} corresponds to O′_{r−k} in the Pyasetskii pairing.

(d) If h_{k} = α_{1}^{∨} + … + α_{k}^{∨} then ⟨e_{k}, h_{k}, f_{k}⟩ is a homogeneous 𝔰𝔩_{2}-subalgebra.

Proof. (d) This is obvious.

(a) By Proposition 2.16 the choice of a maximal string is irrelevant. We take α_{i} = β_{i}, where β_{1}, …, β_{r} is the lower canonical string. First, we need to show that any element in 𝔩_{1} is G-conjugate to one of e_{k}. We argue by the induction on r. Let x ∈ 𝔩_{1}, x ≠ 0. We can write x as

x = ∑_{α∈Δ_{1}} x_{α}e_{α}.

Since 𝔩_{1} is an irreducible G-module, after conjugating x we may assume that x_{β} ≠ 0. Since the G-orbit of x is conical, we may assume that x_{β} = 1. Let Δ̂_{0} = {α ∈ Δ_{0} | (α, β) < 0. Clearly, Δ̂_{0} ⊂ Δ_{0}^{+}. Then

𝔲 = ⟨e_{α} | α ∈ Δ̂_{0}⟩ ⊂ 𝔤

is a unipotent subalgebra. Let U ⊂ G be an unipotent subgroup with 𝔲 = Lie U. Then U acts on 𝔩_{1}. Clearly, for any roots α ∈ Δ̂_{0}, α′ ∈ Δ′_{1} we have α+α′ ∉ Δ by Proposition 2.14 (c). Therefore, U acts trivially on 𝔩′_{1}. Consider the action of U on 𝔩_{1}/𝔩′_{1}. Clearly, the function x_{β} is U-invariant, therefore each U-orbit in 𝔩_{1}/𝔩′_{1} lies on a hyperplane x_{β} = const. For any root α ∈ Δ_{1} such that (α, β) > 0 we have α = β + α̂, where α̂ ∈ Δ̂_{0}. Therefore, the tangent space 𝔲 · (x mod 𝔩′_{1}) is equal to the hyperplane x_{β} = 0. Since any orbit of a linear unipotent group is closed [OV], it follows that the U-orbit of x mod 𝔩′_{1} is the affine hyperplane x_{β} = 1. In particular, x is U-conjugate to an element e_{β} + x′, where x′ ∈ 𝔩′_{1}. By induction, x′ is G′-conjugate to one of e′_{0}, …, e′_{r−1}, where Lie G′ = 𝔩′_{0}. Since G′ · e_{β} = e_{β}, we see that any nonzero element of 𝔩_{1} is G-conjugate to e_{β} + e′_{k} = e_{k+1} for some k = 0, …, r − 1.

To finish the proof of (a) we need to show that all orbits O_{i} are distinct. To prove this it suffices to check that dim 𝔤·e_{i} = dim O_{i} < dim O_{j} = dim 𝔤·e_{j} for i < j. This will follow, in turn, from the inequality

dim Ann(𝔤 · e_{i}) = dim 𝔩_{−1}^{e_{i}} > dim 𝔩_{−1}^{e_{j}} = dim Ann(𝔤 · e_{i}),

where 𝔩_{−1}^{e_{i}} = {x ∈ 𝔩_{−1} | [x, e_{i}] = 0}. Since [f_{i}, 𝔩_{−1}] = 0, it follows from the 𝔰𝔩_{2}-theory that 𝔩_{−1}^{e_{i}} = 𝔩_{−1}^{h_{i}}. But

𝔩_{−1}^{h_{i}} = ⟨e_{−α} | α ∈ Δ^{(i)}⟩,

therefore, dim 𝔩_{−1}^{e_{i}} = #Δ^{(i)}. Since #Δ^{(i)} > #Δ^{(j)}, we see that all orbits O_{i} are distinct.

It is clear from the above that Ann(𝔤 · e_{i}) ∩ Gf_{r−i} is open in Ann(𝔤 · e_{i}), therefore, we have (c).

Finally, to prove (b) we need only to show that O_{i} ⊂ \overline{O_{j}} for i < j. But

lim_{t→−∞} exp t(α_{i+1}^{∨} + … + α_{j}^{∨})e_{j} = e_{i}.

⊓ ⊔

After these case-by-case-independent considerations it is worthy to sort out all examples.

Example 2.18 Consider the short gradings of 𝔩 = 𝔰𝔩_{n+m}. Then

G = {(A, B) ∈ GL_{n} × GL_{m} | det(A) det(B) = 1}.

𝔩_{1} can be identified with ℂ^{n} × ℂ^{m}. There are r = min(n, m) non-zero Gorbits. Namely, O_{i}, i = 1, …, r, is the variety of m × n-matrices of rank i. The projectivization of O_{1} is identified with X = ℙ^{n−1} × ℙ^{m−1} in the Segre embedding. Therefore, the dual variety X^{∗} is equal to the projectivization of the closure \overline{O_{r−1}}, the variety of matrices of rank less than or equal to r − 1. X^{∗} is a hypersurface if and only if n = m, in which case Δ_{X} is the ordinary determinant of a square matrix. Another interesting case is n = 2, m ≥ 2: we see that the Segre embedding of ℙ^{1} × ℙ^{k} is self-dual.

Example 2.19 Consider the short grading of 𝔩 = 𝔰𝔬_{n+2} that corresponds to β being the first simple root. Then G = ℂ^{∗}×SO_{n} and l_{1} = C^{n} with a simplest action. There are two non-zero orbits: the dense one and the self-dual quadric hypersurface Q ⊂ ℂ^{n} preserved by G.

Example 2.20 Consider the short grading of 𝔩 = 𝔰𝔬_{2n} = D_{n} that corresponds to β = α_{n} or β = α_{n−1}. Then G = ℂ^{∗} × SL_{n} acts naturally on 𝔩_{1} = Λ^{2}ℂ^{n}. There are r = [n/2] non-zero orbits, where O_{i}, i = 1, …, r, is the variety of skew–symmetric matrices of rank 2r. The projectivization of O_{1} is identified with X = Gr(2, n) in the Plücker embedding. Therefore, the dual variety X^{∗} is equal to the projectivization of the closure \overline{O_{r−1}}, the variety of matrices of rank less than or equal to 2r − 2. X^{∗} is a hypersurface if and only if n is even, in which case Δ_{X} is the Pfaffian of a skew-symmetric matrix. If n is odd then def X = codim O_{r−1} = 2.

Example 2.21 The short grading of E_{6} gives the following action. G = ℂ^{∗} × SO_{10} and 𝔩_{1} is the half-spinor representation. Except for the zero orbit and the dense orbit there is only one orbit O. In particular, the projectivization of O is smooth and self-dual. It is a spinor variety 𝕊_{5}. It could also be described via Cayley numbers. Let ℂa be the algebra of split Cayley numbers (therefore ℂa = 𝕆 ⊗ ℂ, where O is the real division algebra of octonions). Let u ↦ \overline{u} be the canonical involution in ℂa. Let ℂ^{16} = ℂa ⊕ ℂa have octonionic coordinates u, v. Then the spinor variety S_{5} ⊂ ℙ(ℂ^{16}) is defined by homogeneous equations

u\overline{u} = 0, v\overline{v} = 0, u\overline{v} = 0,

where the last equation is equivalent to 8 complex equations.

Example 2.22 The final example appears from the short grading of E_{7}. Here G = ℂ^{∗} ×E_{6}, and 𝔩_{1} = ℂ^{27} can be identified with the exceptional simple Jordan algebra (the Albert algebra), see [J]. Then E_{6} is the group of norm similarities and ℂ^{∗} acts by homotheties. There are 3 non-zero orbits: the dense one, the cubic hypersurface (defined by the norm in the Jordan algebra), and the closed conical variety with smooth projectivization consisting of elements of rank one, i.e. the exceptional Severi variety (the model of the Cayley projective plane).

40 2. Dual Varieties of Algebraic Group Orbits

3. The Cayley Method for Studying Discriminants

Preliminaries

The remarkable observation due to Cayley [Ca] gives an expression of the discriminant as the determinant of a certain complex of finite-dimensional vector spaces. Of course, complexes were unknown at his time and he was studying resultants rather than discriminants (though he also noticed that resultants and discriminants are, in fact, the equivalent notions). This approach was clarified and developed in the beautiful serie of papers by Gelfand, Kapranov, and Zelevinsky [GKZ1, GKZ2].

3.1 Jet Bundles and Koszul Complexes

Let X be a smooth irreducible algebraic variety with an algebraic line bundle L. We consider the bundle J(L) of first jets of sections of L. By definition, the fiber of J(L) at a point x ∈ X is the quotient of the space of all sections of L near x by the subspace of sections which vanish at x with their first derivatives. In other words, J(L)_{x} = L/I_{x}^{2}L, where I_{x} is the ideal of functions vanishing at x. Thus J(L) is a vector bundle of rank dim X + 1.

To any section f of L, we associate a section j(f) of J(L), called the first jet of f. Namely, the value of j(f) at x is the class of f modulo I_{x}^{2}L. The correspondence f ↦ j(f) is ℂ-linear, but being a differential operator, is not O_{X}-linear.

Let us summarize here without proofs some well-known properties of jet bundles.

Theorem 3.1 ([GKZ2, KS])

(a) For any two line bundles L, M on X, there exists a canonical isomorphism J(L ⊗ M) ≃ J(L) ⊗ M.

(b) Let Ω_{X}^{1} be the sheaf of regular differential 1-forms on X. Then there exists a canonical exact sequence of vector bundles

0 → Ω_{X}^{1} ⊗ L → J(L) → L → 0.  (3.1)

(c) Let X = ℙ(V) be a projective space. Then the jet bundle J(O_{X}(1)) is a trivial vector bundle naturally identified with O_{X} ⊗ V^{∗}

The relevance of jets to dual varieties is as follows. Suppose that X ⊂ ℙ(V) is an irreducible projective variety. We take L = O_{X}(1). Then any f ∈ V^{∗} is a linear function on V and, hence, can be regarded as a global section of L. The following result is an immediate consequence of definitions:

Proposition 3.2 f ∈ V^{∗} represents a point in the dual variety X^{∗} if and only if the section j(f) of a jet bundle J(L) vanishes at some point x ∈ X.

Let E be an algebraic vector bundle of rank r on an irreducible algebraic variety X. For any global section s of E consider the following complexes of sheaves on X, called Koszul complexes.

K_{+}(E, s) = {0 → O_{X} →^{s} E →^{∧s} Λ^{2}E →^{∧s} … →^{∧s} Λ^{r}E → 0} ,

K_{−}(E, s) = {0 → Λ^{r}E^{∗} →^{i_{s}} … →^{i_{s}} Λ^{2}E^{∗} →^{i_{s}} E^{∗} →^{i_{s}} O_{X} → 0} .

The differential in K_{+} is given by the exterior multiplication with s and the differential in K_{−} is given by the contraction with s, that is, by the map Λ^{j}E^{∗} → Λ^{j−1}E^{∗} dual to the map Λ^{j−1}E → Λ^{j}E given by ∧s. We fix gradings in K_{+} and K_{−} by assigning the degree j to Λ^{j}E in K_{+} and by assigning the degree −j to Λ^{j}E^{∗} in K_{−}. Notice that we have an isomorphism of complexes

K_{−}(E, s) ≃ K_{+}(E, s) ⊗ Λ^{r}E^{∗}[r],

where r in brackets means the shift in the grading by r. The following Theorem shows that the cohomology of Koszul complexes ‘represents’ the vanishing set of s.

Theorem 3.3

(a) Koszul complexes K_{+}(E, s) or K_{−}(E, s) are exact if and only if s vanishes nowhere on X (the set of zeros Z(s) is empty).

(b) Suppose that X is smooth and that s vanishes along a smooth subvariety Z(s) of codimension r = rank E, and, moreover, s is transverse to the zero section. Then K_{−}(E, s) has only one non-trivial cohomology sheaf, namely O_{Z(s)} (regarded as a sheaf on X) in highest degree 0. The complex K_{+}(E, s) in this situation has the only non-trivial cohomology sheaf in the highest degree r, and this sheaf is the restriction det E = Λ^{r}E|_{Z(s)} regarded as a sheaf on X.

Sketch of the proof. We shall prove only (a), the proof of (b) follows the same ideas but is more technically involved, see e.g. [Fu1, GH1]. If V is a vector space and v ∈ V is a non-zero vector then the differential in the exterior algebra Λ^{∗}V given by ∧v is exact. Indeed, we can include v in some basis as a first vector, then both the kernel and the image of the differential are spanned by monomials containing v. Therefore, the dual differential i_{v} in Λ^{∗}V^{∗} is also exact. Applying this to our situation we see that if the section s does not vanish anywhere then both Koszul complexes are exact fiberwise and, therefore, are exact as complexes of sheaves. However, if Z(s) is not empty then the cokernel of the last differential in Koszul complexes is not trivial. Indeed, if x ∈ Z(s) then, trivializing E near x, we represent s as a collection of functions (f_{1}, …, f_{r}) ∈ O^{r}. Then the last differential in both complexes has a form O^{r} → O, (u_{1}, …, u_{r}) ↦ ∑ u_{i}f_{i}. If s vanishes at x then all f_{i} = 0 and this map is not surjective. ⊓ ⊔

Let now X ⊂ ℙ(V) be a smooth projective variety. Then we can apply Theorem 3.3 to the jet bundle E = J(L), where L = O_{X}(1). Combining Proposition 3.2 and Theorem 3.3 we get the following

Theorem 3.4 ([GKZ2]) For any f ∈ V^{∗} let j = j(f) denote its first jet. Then a vector f ∈ V^{∗} represents a point in X^{∗} if and only if any of the following complexes of sheaves on X is not exact:

K_{+}(J(L), j) = {0 → O_{X} →^{j} J(L) →^{∧j} Λ^{2}J(L) →^{∧j} … →^{∧j} Λ^{r}J(L) → 0} ,

K_{−}(J(L), j) = {0 → Λ^{r}J(L)^{∗} →^{i_{j}} … →^{i_{j}} Λ^{2}J(L)^{∗} →^{i_{j}} J(L)^{∗} →^{i_{j}} O_{X} → 0} .

3.2 Cayley Determinants of Exact Complexes

The determinants of exact complexes (in the implicit form) were first introduced by Cayley in his paper [Ca] on resultants. A systematic early treatment of this subject was undertaken by Fisher [Fi] whose aim was to give a rigorous proof of Cayley results. In topology determinants of complexes were introduced in 1935 by Reidermeister and Franz [Fra]. They used the word ‘torsion’ for determinant-type invariants constructed. In this section we give only definitions that are necessary for the formulation of results related to dual varieties. More details could be found in [GKZ2, KnMu, Del, Q, RS].

The base field k can be arbitrary. Suppose that V is a finite-dimensional vector space. Then the top-degree component of the exterior algebra Λ^{dim V} V is called the determinant of V, denoted by Det V. If V = 0 then we set Det V = k. It is easy to see that for any exact triple 0 → U → V → W → 0 we have a natural isomorphism Det V ≃ Det U ⊗ Det W.

Suppose now that V = V_{0} ⊕ V_{1} is a finite-dimensional supervector space. Then, by definition, Det V is set to be Det V_{0} ⊗ (Det V_{1})^{∗}. Once again, for any exact triple of supervector spaces 0 → U → V → W → 0 we have Det V ≃ Det U ⊗ Det W. For any supervector space V we denote by Ṽ the new supervector space given by Ṽ_{0} = V_{1}, Ṽ_{1} = V_{0}. Clearly, we have a natural isomorphism

Det Ṽ = (Det V)^{∗}.  (3.2)

Now let (V, ∂) be a finite-dimensional supervector space with a differential ∂ such that ∂V_{0} ⊂ V_{1}, ∂V_{1} ⊂ V_{0}, ∂^{2} = 0. Then Ker ∂, Im ∂, and the cohomology space H(V) = Ker ∂/ Im ∂ are again supervector spaces. We claim that there exists a natural isomorphism

Det V ≃ Det H(V).  (3.3)

Indeed, from the exact sequence

0 → Ker ∂ → V →^{∂} Ĩm ∂ → 0

we see that Det V ≃ Det(Ker ∂) ⊗ Det(Im ∂)^{∗}. From the exact sequence

0 → Im ∂ → Ker ∂ → H(V) → 0

we get that Det H(V) ≃ Det(Ker ∂) ⊗ Det(Im ∂)^{∗}. Therefore, we have (3.3). In particular, if ∂ is exact, H(V) = 0, then we have a natural isomorphism

Det V ≃ k.  (3.4)

Let us fix some bases {e_{1}, …, e_{dim V_{0}}} in V_{0} and {e′_{1}, …, e′_{dim V_{1}}} in V_{1}. Let {f′_{1}, …, f′_{dim V_{1}}} be a dual basis in V_{1}^{∗}. Then we have a basis vector

e_{1} ∧ … ∧ e_{dim V_{0}} ⊗ f′_{1} ∧ … ∧ f′_{dim V_{1}} ∈ Det V.

Therefore, if (V, ∂, e) is a based supervector space with an exact differential then by (3.4) we get a number det(V, ∂, e) called the Cayley determinant of a based supervector space with an exact differential. If we fix other bases {ẽ_{1}, …, ẽ_{dim V_{0}}} in V_{0} and {ẽ′_{1}, …, ẽ′_{dim V_{1}}} in V_{1} then, clearly,

det(V, ∂, ẽ) = det A_{0}(det A_{1})^{−1} det(V, ∂, e),  (3.5)

where (A_{0}, A_{1}) ∈ GL(V_{0}) × GL(V_{1}) are transition matrices from bases e to bases ẽ. One upshot of this is the fact that if bases e and ẽ are equivalent over some subfield k_{0} ⊂ k then the Cayley determinants with respect to these bases are equal up to a non-zero multiple from k_{0}. Since all isomorphisms are natural so far, the homogeneity condition (3.5) can be reformulated as follows. For any based supervector space (V, ∂, e) with an exact differential and any (A_{0}, A_{1}) ∈ GL(V_{0}) × GL(V_{1}) we can consider a new based supervector space (V, A · ∂, e) with the exact differential A · ∂ given by

A · ∂|_{V_{0}} = A_{1} · ∂ · A_{0}^{−1}, A · ∂|_{V_{1}} = A_{0} · ∂ · A_{1}^{−1}.

Then we have

det(V, A · ∂, e) = (det A_{0})^{−1} det A_{1} det(V, ∂, e).

In particular, it is easy to see that the following formula is valid:

det(V, λ∂, e) = λ^{dim(Ker ∂)_{1}−dim(Ker ∂)_{0}} det(V, ∂, e).  (3.6)

In the matrix form the Cayley determinant can be calculated as follows. Suppose that (V, ∂, e) is a based supervector space with an exact differential,

dim V_{0} = dim Ker ∂_{0} + dim Im ∂_{0} = dim Im ∂_{1} + dim Ker ∂_{1} = dim V_{1} = n.

Therefore, both ∂_{0} and ∂_{1} are represented by (n×n)-matrices D_{0} and D_{1}. For any subsets I, J ⊂ B = {1, …, n} we denote by D_{0}^{IJ} and D_{1}^{IJ} the submatrices of D_{0} and D_{1} formed by columns indexed by I and rows indexed by J. Then it is easy to see that there exist subsets I_{0}, I_{1} ⊂ B such that submatrices D_{0}^{B\I_{0},I_{1}}, D_{1}^{B\I_{1},I_{0}} are invertible. In particular,

#I_{1} = rk ∂_{0}, #I_{0} = rk ∂_{1} = n − #I_{1}.

The formula (3.5) implies that

det(V, ∂, e) = det D_{0}^{B\I_{0},I_{1}}(det D_{1}^{B\I_{1},I_{0}})^{−1}.  (3.7)

This formula gives an explicit matrix description of the Cayley determinant. Now we consider a finite complex of finite-dimensional vector spaces

… →^{∂_{i−1}} V^{i} →^{∂_{i}} V^{i+1} →^{∂_{i+1}} ….

Then we can define the finite-dimensional supervector space V = V_{0} ⊕ V_{1},

V_{0} = ⊕_{i≡0 mod 2} V^{i}, V_{1} = ⊕_{i≡1 mod 2} V^{i},

with an induced differential ∂. In particular, all previous considerations are valid. Therefore, if the complex (V^{•}, ∂) is exact and there are some fixed bases {e_{1}^{i}, …, e_{dim V^{i}}^{i}} in each component V^{i} then we have the corresponding Cayley determinant det(V^{•}, ∂, e) ∈ k^{∗}. For example, if L and M are based vector spaces and A : L → M is an invertible operator then the complex 0 → L →^{A} M → 0 is exact and the corresponding Cayley determinant is equal to det A (if L is located in the even degree of the complex).

In fact, Cayley in [Ca] has found some matrix representation of his determinant. In order to give his formula we shall write our complex in the explicit coordinate form

V^{•} = {0 → k^{B_{0}} →^{D_{0}} k^{B_{1}} →^{D_{1}} → … →^{D_{r−1}} k^{B_{r}} → 0},  (3.8)

where for simplicity we assume that the non-zero terms of the complex are located in degrees between 0 and r > 0. In general, if V^{•}[m] is the same complex as V^{•} but with a grading shifted by m, then by (3.2) we have det(V^{•}[m], ∂, e) = (det(V^{•}[m], ∂, e))^{(−1)^{m}} In the formula (3.8) we may suppose that all B_{i} are some finite sets. Therefore, D_{i} is a matrix with columns indexed by B_{i} and rows indexed by B_{i+1}. For any subsets X ⊂ B_{i}, Y ⊂ B_{i+1} we denote by (D_{i})_{XY} the submatrix in D_{i} with columns from X and rows from Y.

A collection of subsets I_{i} ⊂ B_{i} is called admissible if I_{0} = ∅, I_{r} = B_{r}, for any i = 0, …, r − 1 we have #(B_{i} \ I_{i}) = #I_{i+1}, and the submatrix (D_{i})_{B_{i}\I_{i},I_{i+1}} is invertible. The following theorem easily follows from (3.7).

Theorem 3.5 ([GKZ2])

(a) Admissible collections exist. For any admissible collection we have

#I_{i} = ∑_{j=0}^{i−1}(−1)^{i−1−j}#B_{j}.

(b) Let {I_{i}} be an admissible collection. Denote by Δ_{i} the determinant of the matrix (D_{i})_{B_{i}\I_{i},I_{i+1}}. Then

det(V^{•}, D, B) = ∏_{i=0}^{r−1} Δ_{i}^{(−1)^{i}}.

In particular, this Theorem (or formula (3.6)) implies

Corollary 3.6 For any based exact complex (V^{•}, d, e) we have

det(V^{•}, λd, e) = λ^{∑_{i}(−1)^{i+1}·i·dim V^{i}} det(V^{•}, d, e).

3.3 Discriminant Complexes

Let X ⊂ ℙ(V) be a smooth projective variety. We denote L = O_{X}(1), so any f ∈ V^{∗} can be regarded as a section of L. Let M be another line bundle on X. We define discriminant complexes C_{+}^{∗}(X, M) and C_{−}^{∗}(X, M) as complexes of global sections of Koszul complexes K_{+}(J(L), j(f)) and K_{−}(J(L), j(f)) tensored by M. More precisely,

C_{+}^{i}(X, M) = H^{0}(X, Λ^{i}J(L) ⊗ M),
C_{−}^{i}(X, M) = H^{0}(X, Λ^{−i}J(L)^{∗} ⊗ M).

Thus the terms of discriminant complexes are fixed and the differentials depend on f ∈ V^{∗}. We shall denote this differential by ∂_{f}. By Theorem 3.4 a vector f ∈ V^{∗} represents a point in X^{∗} if and only if any of two Koszul complexes is not exact. We want to get the same condition but for discriminant complexes, which are complexes of finite-dimensional vector spaces instead of complexes of sheaves. However, the exactness of a complex of sheaves does not necessarily imply the exactness of the corresponding complex of global sections. The obstruction to this is given by the higher cohomology of the sheaves of the complex.

Definition 3.7 The discriminant complexes are called stably twisted if all terms of the corresponding Koszul complexes have no higher cohomology.

For example, suppose that M is any ample line bundle on X. Ampleness is equivalent to the fact that for any coherent sheaf F on X, the sheaves F ⊗ M^{⊗n} have no higher cohomology for n ≫ 0. Therefore, for sufficiently big n the discriminant complexes C_{+}^{∗}(X, M^{⊗n}) and C_{−}^{∗}(X, M^{⊗n}) will be stably twisted. In this situation we have the following Theorem that immediately follows from Theorem 3.4 and ‘abstract de Rham theorem’ (see e.g.[GH1]).

Theorem 3.8 ([GKZ2]) Suppose that the discriminant complex C_{+}^{∗}(X, M) (resp. C_{−}^{∗}(X, M)) is stably twisted. Let f ∈ V^{∗} be such that its projectivization does not belong to the dual variety X^{∗} ⊂ ℙ(V^{∗}). Then the complex (C_{+}^{∗}(X, M), ∂_{f}) (resp. (C_{−}^{∗}(X, M), ∂_{f})) is exact.

Remarkably, it turns out that much more is true. Suppose that the discriminant complex C_{+}^{∗}(X, M) (or C_{−}^{∗}(X, M)) is stably twisted. Then for a generic f ∈ V^{∗} the complex C_{+}^{∗}(X, M) (or C_{−}^{∗}(X, M)) is exact by the previous Theorem. Let f_{1}, …, f_{n} be a basis of V^{∗}. Let ℂ(x_{1}, …, x_{n}) = ℂ(V^{∗}) be a field of rational functions on V^{∗}. Consider the complex C_{+}^{∗}(X, M) ⊗ ℂ(V^{∗}) (or C_{−}^{∗}(X, M)⊗ℂ(V^{∗})) with the differential given by ∂ = ∑ x_{i}∂_{f_{i}}. Then this complex is exact, since its generic specialization is exact. We fix some bases over ℂ in each component C_{+}^{i}(X, M) (or C_{−}^{−i}(X, M)) and consider them as bases over ℂ(V^{∗}) in each C_{+}^{i}(X, M) ⊗ ℂ(V^{∗}) (or C_{−}^{−i}(X, M) ⊗ ℂ(V^{∗})). Then we can calculate the Cayley determinant Δ_{X,M}^{+} (or Δ_{X,M}^{−}) of the complex C_{+}^{∗}(X, M) ⊗ ℂ(V^{∗}) (or C_{−}^{∗}(X, M) ⊗ ℂ(V^{∗})), which will be the non-zero element of the field of rational functions ℂ(V^{∗}). Clearly the Cayley determinants depend only on the discriminant complexes (up to a scalar multiple) and do not depend on the choice of bases. The following Theorem shows that the Cayley determinant coincides with the discriminant.

Theorem 3.9 ([GKZ2]) If the discriminant complex C_{−}^{∗}(X, M) is stably twisted then, up to a non-zero scalar, we have

Δ_{X,M}^{−} = Δ_{X},

where Δ_{X} is the discriminant of X. If C_{+}^{∗}(X, M) is stably twisted then, up to a non-zero scalar, we have

(Δ_{X,M}^{+})^{(−1)^{dim X+1}} = Δ_{X}.

The proof of this Theorem involves derived categories and Cayley determinants for complexes over arbitrary Noetherian integral domains, see [GKZ2]. Let us consider several examples.

Sylvester Formula. Let X = ℙ^{1} = ℙ(ℂ^{2}) be a projective line embedded into ℙ(S^{d}ℂ^{2}) via the Veronese embedding. The space V^{∗} = (S^{d}ℂ^{2})^{∗} is the space of binary forms

f(x_{0}, x_{1}) = a_{0}x_{0}^{d} + a_{1}x_{0}^{d−1}x_{1} + … + a_{d}x_{1}^{d}

and the discriminant Δ_{X}(f) is the classical discriminant of this binary form. It is quite easy to implement Theorem 3.9 in this case. We take a twisting bundle M = O_{ℙ^{1}}(2d − 3). The discriminant complex C_{−}^{∗}(X, M) has only two non-zero terms and, therefore, the Cayley determinant is reduced to a determinant of a square matrix. More precisely, the Cayley determinant in this case is equal to the determinant of the linear map

∂_{f} : S^{d−2}ℂ^{2} ⊕ S^{d−2}ℂ^{2} → S^{2d−3}ℂ^{2},

given by

∂_{f}(u, v) = \frac{∂f}{∂x_{0}} u + \frac{∂f}{∂x_{1}} v.

The final formula coincides with the classical Sylvester formula for the discriminant of a binary form (up to a scalar):

Δ_{X}(f) = \frac{(−1)^{d−1}}{d^{d−2}} D,

where D is the determinant of the following matrix

| \begin{matrix}a_{1} & 2a_{2} & … & (d − 1)a_{d−1} & da_{d} & … & 0 \\ 0 & a_{1} & … & (d − 2)a_{d−2} & (d − 1)a_{d−1} & … & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & a_{1} & 2a_{2} & … & da_{d} \\ da_{0} & (d − 1)a_{1} & … & 2a_{d−2} & a_{d−1} & … & 0 \\ 0 & da_{0} & … & 3a_{d−3} & 2a_{d−2} & … & 0 \\ ⋮ & ⋮ & ... & ⋮ & ⋮ & ... & ⋮ \\ 0 & 0 & … & da_{0} & (d − 1)a_{1} & … & a_{d−1} \end{matrix} | .

The Dual of an Algebraic Curve. Let X ⊂ ℙ(V) be a smooth algebraic curve of degree d and genus g. We take the twisting bundle M to be a generic line bundle on X of degree 2d + 3g − 3. The discriminant complex C_{−}^{∗}(X, M) has only two non-zero terms and, therefore, the Cayley determinant is given by the determinant of a square matrix. The size of this matrix (and hence the degree of the dual variety X^{∗}) is equal to dim H^{0}(X, M). The latter is equal to 2d + 2g − 2 by the Riemann-Roch theorem.

Discriminant Spectral Sequences. One disadvantage of Theorem 3.9 is the restricting condition on the twisting line bundle M. It is not always convenient to make the discriminant complexes C_{±}^{•} stably twisted. Another possibility is to take into account the higher cohomology as well. The standard tool for this is the spectral sequence. Recall [GH1] that if F^{•} is a finite complex of sheaves on a topological space X, then one can define the hypercohomology groups H^{i}(X, F^{•}). To define them, consider the complexes of Abelian groups C_{j}^{•} calculating the cohomology of every individual F^{j} (for example, the C̆ech complexes with respect to an appropriate open covering of X). The differentials F^{i} → F^{i+1} make this collection of complexes into a double complex C^{••}. The hypercohomology groups H^{i}(X, F^{•}) are the cohomology groups of its total complex. In particular, we have the spectral sequence of the double complex C^{••}

E_{1}^{pq} = H^{q}(X, F^{p}) ⇒ H^{p+q}(X, F^{•}).

The first differential d_{1} is induced by the differential in F^{•}. In particular, the complex of global sections

… → H^{0}(X, F^{i}) → H^{0}(X, F^{i+1}) → …

is just the bottom row of the term E_{1}.

If the complex of sheaves F^{•} is exact then all hypercohomology groups H^{i}(X, F^{•}) vanish. Indeed, the hypercohomology can be calculated using another spectral sequence ‘dual’ to the first:

′E_{2}^{pq} = H^{q}(X, H^{p}(F^{•})) ⇒ H^{p+q}(X, F^{•}),

where H^{p}(F^{•}) is the p-th cohomology sheaf of F^{•}

H^{p}(F^{•}) = \frac{Ker\{F^{p} →^{d} F^{p+1}\}}{Im\{F^{p−1} →^{d} F^{p}\}}.

Now let X ⊂ ℙ(V) be a smooth projective variety, L = O_{X}(1). Let M be any line bundle on X. We define discriminant spectral sequences C_{r,±}^{pq}(X, M, f) to be the spectral sequences of complexes K_{±}(J(L), j(f)) tensored by M. Then the following theorem is a consequence of the discussion above and Theorem 3.4.

Theorem 3.10 ([GKZ2]) Suppose that the projectivization of f ∈ V^{∗} does not belong to the dual variety X^{∗}. Then the discriminant spectral sequences C_{r,±}^{pq}(X, M, f) are exact, i.e. they converge to zero.

Moreover, it is possible to define the Cayley determinants of exact spectral sequences and to prove an analogue of Theorem 3.9, see [GKZ2] for details.

Weyman’s Bigraded Complexes The calculation of the determinant of a spectral sequence with many non-trivial terms may be very involved. J. Weyman [We] has suggested a procedure that replaces the discriminant spectral sequence by a bigraded complex which incorporates all higher differentials at once. Bigraded complex is, by definition, a complex (C^{•}, ∂) of vector spaces in which each term C^{i} is equipped with additional grading C^{i} = ⊕_{p+q=i}C^{pq}. The underlying complex (C^{•}, ∂) may then be called a total complex of the bigraded complex C^{••}. The differential ∂ in C^{•} is decomposed into a sum ∂ = ∑_{r} ∂_{r}, where ∂_{r} is bihomogeneous of degree (r, 1 − r). We shall consider only bigraded complexes with ∂_{r} = 0 for r < 0.

Now let X ⊂ ℙ(V) be a smooth projective variety, L = O_{X}(1). Let M be any line bundle on X.

Theorem 3.11 ([We]) There exists a bigraded complex (C^{••}, ∂_{f}) with

C^{pq} = H^{q} (X, Λ^{−p}J(L)^{∗} ⊗ M)

and the differential ∂_{f} depends polynomially on f ∈ V^{∗}. This complex has the following properties:

– The differential ∂_{f} has the form ∂_{f} = ∑_{r≥1} ∂_{r,f}, where ∂_{r,f} has bidegree (r, 1 − r) and its matrix elements are homogeneous polynomials of degree r in coefficients of f. Moreover, ∂_{1,f} is induced by the differential in the Koszul complex K_{−}(J(L), j(f)).

– The Cayley determinant of the total complex (C^{•}, ∂_{f}) of C^{••} equals, up to a scalar factor, the discriminant Δ_{X}.

4. Resultants and Schemes of Zeros

Preliminaries

Classically, discriminants were studied together with resultants. Resultants and their generalizations appear while looking at vector bundles: they detect their global sections that have “correct” schemes of zeros. Some results in this direction are reviewed in this chapter.

4.1 Ample Vector Bundles

Classically, a line bundle L over an algebraic variety X is said to be ample if the map X → ℙ^{N} associated to the sections of some power L^{n} gives an imbedding of X. This notion has been extended to vector bundles as well by Hartshorne [Ha3]. Let E be a vector bundle on X. We consider the variety X_{E} = ℙ(E^{∗}), the projectivization of the bundle E^{∗}. There is a projection p : E_{X} → X whose fibers are projectivizations of fibers of E^{∗}, and a natural projection π : E^{∗} \ X → X_{E}, where X is embedded into the total space of E^{∗} as the zero section. We denote by ξ(E) the tautological line bundle on X_{E} defined as follows. For open U ⊂ X_{E}, a section of ξ(E) over U is a regular function on π^{−1}(U) which is homogeneous of degree 1 with respect to dilations of E^{∗}. The restriction of ξ(E) to every fiber p^{−1}(X) = ℙ(E_{x}^{∗}) is the tautological line bundle O(1) of the projective space ℙ(E_{x}^{∗}).

Definition 4.1 A vector bundle E on X is called ample if ξ(E) is an ample line bundle on X_{E}.

If E is itself a line bundle then X_{E} = X, ξ(E) = E, and this definition gives nothing new. Basic properties and criteria for ampleness were obtained in [Ha3]. For example, the Grothendieck definition of ample line bundles can be transported to vector bundles as well. Namely, E is ample on X if and only if for every coherent sheaf F, there is an integer n_{0} > 0, such that for every n ≥ n_{0}, the sheaf F ⊗ S^{n}(E) (where S^{n}(E) is the n-th symmetric power of E) is generated as an O_{X}-module by its global sections. Using this description it is easy to see, for example, that the direct sum of ample vector bundles is ample, etc.

The following theorem is an easy application of ample vector bundles properties.

Theorem 4.2 ([E1]) Assume that X is a nonlinear smooth projective variety in ℙ^{N}. If X is a complete intersection then X^{∗} is a hypersurface.

Proof. We may assume that X is nondegenerate. Let dim X = n. Then X is scheme-theoretic intersection of N − n hypersurfaces H_{1}, …, H_{N−n} such that deg H_{k} = d_{k} ≥ 2. Therefore the normal bundle N_{X}ℙ^{N} is equal to O_{X}(d_{1}) ⊕ … ⊕ O_{X}(d_{N−n}). It follows that

N_{X}ℙ^{N}(−1) = O_{X}(d_{1} − 1) ⊕ … ⊕ O_{X}(d_{N−n} − 1)

is an ample vector bundle. Let I_{X} = ℙ(N_{X}^{∗}ℙ^{N}(1)) ⊂ ℙ^{N} × (ℙ^{N})^{∗} be the conormal variety and let pr : I_{X} → X^{∗} be the projection. Then pr^{∗} O_{X^{∗}}(1) is the tautological line bundle on ℙ(N_{X}^{∗}ℙ^{N}(1)). Since N_{X}ℙ^{N}(−1) is ample, it follows that pr is a finite morphism. Therefore,

dim X^{∗} = dim I_{X} = N − 1.
⊓ ⊔

Sommese [S1] generalized the notion of ampleness to the following.

Definition 4.3 A vector bundle E is called k-ample if ξ(E)^{n} is spanned by global sections for some n > 0 and the induced map X_{E} → ℙ^{N} has at most k-dimensional fibers.

It can be verified that the number k is independent of the integer n for which ξ(E)^{n} is spanned and a vector bundle is 0-ample if and only if it is ample. This number k is called the ampleness of E, denoted by a(E). It can be shown that 0 ≤ a(E) ≤ dim X. If ξ(E)^{n} is not spanned for any n > 0, we define a(E) = dim X.

The ampleness of homogeneous vector bundles on flag varieties was described combinatorially by Snow [Sn1] inspired by an earlier work of Goldstein [Go] who determined the ampleness for tangent bundles on flag varieties.

4.2 Resultants

Classically, discriminants were studied in conjunction with resultants. Let X be a smooth irreducible projective variety and let E be a vector bundle on X of rank k = dim X + 1. Set V = H^{0}(X, E). We shall assume that E is very ample, i.e. ξ(E) is a very ample line bundle on X_{E}. In particular, ξ(E) (and hence E) is generated by global sections. Notice that H^{0}(X, E) = H^{0}(X_{E}, ξ(E)), therefore ξ(E) embeds X_{E} in ℙ(V^{∗}).

Definition 4.4 The resultant variety ∇ ⊂ ℙ(V) is the set of all sections vanishing at some point x ∈ X.

Example 4.5 Suppose that X = ℙ^{k−1} = ℙ(ℂ^{k}) and

E = O(d_{1}) ⊕ … ⊕ O(d_{k}).

Then V = S^{d_{1}}(ℂ^{k})^{∗} ⊕ … ⊕ S^{d_{k}}(ℂ^{k})^{∗} and ∇ is the classical resultant variety parametrizing k-tuples of homogeneous forms on ℂ^{k} of degrees d_{1}, …, d_{k} having a common non-zero root.

The assertion (b) of the following Theorem is sometimes called the Cayley trick who first noticed that the resultant can be written as a discriminant.

Theorem 4.6 ([GKZ2])

(a) ∇ is an irreducible hypersurface of degree ∫_{X} c_{k−1}(E).

(b) ∇ is projectively dual to X_{E}.

Proof. ξ(E) embeds X_{E} in ℙ(V^{∗}) in such a way that all fibers become projective subspaces of dimension k − 1. Geometrically, ∇ parametrizes hyperplanes in ℙ(V^{∗}) that contain at least one fiber of X_{E}. Since the embedded tangent space to any point of X_{E} contains the fiber through it, it follows that ∇ ⊃ X_{E}^{∗}. On the contrary, suppose that v ∈ ∇. Then the corresponding section s ∈ H^{0}(X, E) vanishes at some point x ∈ X. Therefore, s defines a linear map ds : T_{x}X → T_{x}E = T_{x}X ⊕ E_{x}. Let pr : T_{x}E → E_{x} be the projection. Since dim X = dim E − 1, pr ∘ds(T_{x}X) is contained in some hyperplane H ⊂ E_{x}. An easy local calculation shows that the hyperplane in ℙ(V^{∗}) corresponding to v contains the embedded tangent space to X_{E} at the point corresponding to H. Therefore, ∇ = X_{E}^{∗}.

Now let us prove that ∇ is a hypersurface. An easy dimension count shows that it is sufficient to prove that a generic section s of H^{0}(X, E) vanishing at x ∈ X does not vanish at other points. In other words, a generic hyperplane in P = ℙ(V^{∗}) containing some fiber F of E_{X} does not contain other fibers. Consider the linear projection π_{F} : P \ F → P/F. If F′ is any fiber of X_{E} distinct from F then π_{F} (F′) is a projective subspace of dimension dim F′ = k − 1. Therefore, images of fibers of X_{E} distinct from F form at most (k − 1)-dimensional family of projective subspaces of dimension k − 1. An easy dimension count shows that there exists a hyperplane H ⊂ P/F that does not contain any projective subspace from this family. The preimage of this hyperplane in P contains only one fiber F.

It remains to calculate the degree of ∇. To find it, we should take two generic sections s_{1} and s_{2} of V and find the number of values λ ∈ ℂ such that s_{1} + λs_{2} vanishes at some x ∈ X. In other words, we need to find the number of points (with multiplicities) of a zero-dimensional cycle Z ⊂ X consisting of all points x ∈ X where s_{1} and s_{2} are linearly dependent. But this is exactly the geometric definition of c_{k−1}(E), see [Fu1]. ⊓ ⊔

The same argument also proves the following result from the folklore. Suppose that X ⊂ ℙ^{N} is a scroll, i.e. a projectivization of a vector bundle E over a smooth variety Y such that all fibers are embedded linearly. Assume that dim E > dim Y. Then

def X = dim E − dim Y − 1

and X^{∗} parametrizes hyperplanes in ℙ^{N} that contain at least one fiber of the projective bundle X → Y.

The proof of the following theorem can be found in [E2]:

Theorem 4.7 ([E2]) If X is a smooth n-dimensional projective variety in ℙ^{N} and def X = k ≥ n/2, then X is a projective bundle X ≃ ℙ_{Y} (F), where F is a vector bundle of rank (n+k +2)/2 on a smooth (n−k)/2-dimensional variety Y and the fibers are embedded linearly.

4.3 Zeros of Generic Global Sections

Both resultants and discriminants are related to the following general construction. Suppose that X is a smooth projective variety and E is a vector bundle on X generated by global sections. Let Z(s) denote the scheme of zeros of any global section s ∈ H^{0}(X, E). One might expect that for generic s the scheme Z(s) is a smooth variety of codimension dim E. Then we can define the degeneration variety D ⊂ H^{0}(X, E) parametrizing all global sections s such that Z(s) is not smooth of expected codimension. For example, if E is a very ample line bundle then D is a cone over the dual variety. If E is a very ample vector bundle and dim E = dim X + 1 then D is the resultant variety. If E is a very ample vector bundle and dim E = dim X then the corresponding homogeneous polynomial can be called Bezoutian. Indeed, if X = ℙ^{n} and E = O(d_{1}) ⊕ … ⊕ O(d_{n}) then D parametrizes sets of homogeneous forms of degrees d_{1}, …, d_{n} such that the Bezout theorem is not applicable.

In general, in order to make the theory consistent, it is necessary to impose very strong conditions on the vector bundle E. These conditions are not always satisfied even in the case of homogeneous vector bundles on flag varieties. Even the question whether or not a generic zero scheme is non-empty can be quite difficult.

Assume that G is a connected reductive group, T is a fixed maximal torus, B is a fixed Borel subgroup, T ⊂ B ⊂ G, B_{−} is the opposite Borel subgroup, P is a parabolic subgroup containing B_{−}, X(T) is the lattice of characters of T, and λ ∈ X(T) is the dominant weight. Consider the homogeneous vector bundle L_{λ} = G ×_{P} U_{λ} over G/P, where U_{λ} is the irreducible P -module with highest weight λ. See Section 2.1 for further details. By the Borel–Weil–Bott theorem (see [Bot]), V_{λ} = H^{0}(G/P, L_{λ}) is an irreducible G-module with highest weight λ.

Theorem 4.8 Let s ∈ V_{λ} be a generic global section. Then (a) If dim U_{λ} > dim G/P, the scheme of zeros Z_{s} is empty.

(b) If dim U_{λ} ≤ dim G/P, either Z_{s} is empty or s intersects the zero section of L_{λ} transversally and Z_{s} is a smooth unmixed subvariety of expected codimension dim U_{λ}.

(c) If dim U_{λ} = dim G/P, the geometric number of points in Z_{s} is equal to the top Chern class of L_{λ}.

Proof. (a) is obvious and follows by an easy dimension count. (b) and (c) follow from the fact that L_{λ} is generated by global sections. Let us recall this argument here. Let dim U_{λ} ≤ dim G/P, and assume that every global section has a zero. We have to prove that a generic global section s intersects the zero section of L_{λ} transversally. This will imply, in particular, that Z_{s} is a smooth unmixed subvariety of codimension dim U_{λ}. For simplicity we suppress the index λ. Consider the incidence variety

Z ⊂ G/P × V, Z = {(x, s) | x ∈ (Z_{s})_{red}}.

Since G/P is homogeneous and Z is invariant, it follows that Z is obtained by spreading the fiber Z_{e} = {s ∈ V | s(eP) = 0} by the group G. Since U is irreducible, we have dim Z_{e} = dim V − dim U. Hence, Z is a smooth irreducible subvariety of dimension dim V + dim G/P − dim U.

Let π : Z → V be the restriction to Z of the projection of G/P × V on the second summand. By assumption, π is a surjection. By Sard’s lemma for algebraic varieties (see [Mum1]), for a generic point s ∈ V and any point (x, s) in π^{−1}s the differential dπ_{(x,s)} is surjective. We claim that s has the transversal intersection with the zero section. Indeed, Z can be regarded as a subbundle of the trivial bundle G/P × V. Then L = (G/P × V)/Z and the zero section of L is identified with Z mod Z. The section s is identified with (G/P × {s}) mod Z. Hence, it is sufficient to prove that G/P × {s} is transversal to Z, which is equivalent to the following claim: dπ_{(x,s)} is surjective for all (x, s) ∈ Z. Now statement (c) of the theorem follows from the standard intersection theory (see [Fu1]). ⊓ ⊔

Example 4.9 It should be noted that Theorem 4.8 cannot be strengthened to the point where the non-emptiness and the irreducibility of the scheme of zeros in Theorem 4.8 could be established apriori, as the following example shows. Consider the vector bundle S^{2}𝕊^{∗} on Gr(k, 2n). The dimension of a

fiber does not exceed the dimension of the Grassmanian as k ≤ \frac{4n − 1}{3}, but

a generic section (that is, a non-degenerate quadratic form in ℂ^{2n}) has a zero (that is, a k-dimensional isotropic subspace) only if k ≤ n. For k = n

the scheme of zeros is a reducible variety of dimension \frac{n(n − 1)}{2} with two

irreducible components (spinor varieties) that correspond to two families of maximal isotropic subspaces on an even-dimensional quadric.

Consider the P -submodule M_{λ} = {s ∈ H^{0}(G/P, L_{λ}) | s(P) = 0} in V_{λ}. Here s(P) is a germ of s in the point of G/P corresponding to P. It is easy to see that M_{λ} can be characterized as the unique maximal proper P - submodule of V_{λ}. Clearly V_{λ}/M_{λ} ≃ U_{λ} as P -modules. Consider the map Ψ : G × M_{λ} → V_{λ}, Ψ(g, v) = gv. Then generic global sections of L have zeros iff Ψ is dominant iff the differential of Ψ at a generic point is surjective. We write 𝔤 and 𝔭 for Lie algebras of G and P, respectively. The natural “orbital” map 𝔤 × V_{λ} → V_{λ} defines the P -equivariant map ψ : 𝔤/𝔭 × M_{λ} → U_{λ}. Then Ψ is dominant iff ψ(·, x) is surjective at a generic point x ∈ M_{λ}. Let Z_{λ} = ψ^{−1}(0)_{red} ⊂ 𝔤/𝔭 × M_{λ} be the incidence variety. Then the following proposition follows from an easy dimension count:

Proposition 4.10 Suppose that

dim Z_{λ} = dim 𝔤/𝔭 + dim M_{λ} − dim U_{λ}.

Then generic global sections of L_{λ} have zeros.

Denote by π the restriction on Z_{λ} of the projection of 𝔤/𝔭×M_{λ} to the first factor. The variety Z_{λ} could be rather complicated. Say, it is not irreducible in general. We shall use the fact that fibers π^{−1}(x) are linear spaces. So consider the function l_{λ}(x) = dim π^{−1}(x), x ∈ 𝔤/𝔭.

Conjecture 4.11 There exists an algebraic stratification 𝔤/𝔭 = ⊔_{i=1}^{r} X_{i} such

that for any λ the function l_{λ}(x) is constant along each X_{i}:

l_{λ}(x) = l_{λ}^{i}, x ∈ X_{i}.

If such stratification exists then

dim Z_{λ} = max_{i}(dim X_{i} + l_{λ}^{i}).  (4.1)

If P acts on 𝔤/𝔭 with a finite number of orbits then we can take these orbits as X_{i} (since ψ is P -equivariant). Unfortunately, this class of parabolic subgroups is very small. The dual P -module (𝔤/𝔭)^{∗} is isomorphic to the representation of P on the unipotent radical 𝔭_{n} of 𝔭, hence by Pyasetskii Theorem 2.9 the action P : 𝔤/𝔭 has a finite number of orbits iff the action P : 𝔭_{n} satisfies this property. Such actions were studied in [PR], the complete classification for classical groups was obtained in [HRo]. But even in these cases the problem of the complete description of the orbital decomposition seems very messy. In the next section we shall introduce another class of parabolic subgroups that satisfy the Conjecture 4.11.

The most wonderful class of parabolic subgroups that will fit all our needs is the class of parabolic subgroups P with abelian unipotent radical 𝔭_{n}, see Section 2.3. So from now on P will be a parabolic subgroup with aura. Let P^{−} be an opposite parabolic subgroup, let L = P ∩ P^{−} be a Levi subgroup with Lie algebra 𝔩, let 𝔭_{n}^{−} be a unipotent radical of 𝔭^{−}. Then 𝔭_{n} is abelian iff the decomposition 𝔤 = 𝔭_{n}^{−} ⊕ 𝔩 ⊕ 𝔭_{n} is a ℤ-grading: 𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}. Notice that 𝔤/𝔭 is isomorphic to 𝔭_{n}^{−} as a L-module. It is well-known that the action of L on 𝔭_{n}^{−} has a finite number of orbits in this case. Take the orbital decomposition 𝔭_{n}^{−} = ∪_{i=1}^{r} Lf_{i}. For its detailed description see Theorem 2.17.

Then the equality (4.1) takes a form

dim Z_{λ} = max_{i}(dim Lf_{i} + l_{λ}(f_{i})).  (4.2)

It remains to calculate l_{λ}(f_{i}). If f_{i} = 0 then l_{λ}(f_{i}) = dim M_{λ}. In the opposite case we can include f_{i} in a 𝔰𝔩_{2}-triple ⟨f_{i}, h_{i}, e_{i}⟩. Evidently we can assume that h_{i} ∈ 𝔩, e_{i} ∈ 𝔭_{n}. These h_{i}’s were also written down explicitly in Theorem 2.17. We shall use the following easy lemma from the 𝔰𝔩_{2}-theory:

Lemma 4.12 Let V be a finite-dimensional 𝔰𝔩_{2}-module, 𝔰𝔩_{2} = ⟨f, h, e⟩. Let 𝔟 = ⟨h, e⟩. Suppose that M ⊂ V is a 𝔟-submodule such that eV ⊂ M. Then

dim Coker(M ⊂ V →^{f} V → V/M) = dim(V/M)^{h},

where (V/M)^{h} = {v ∈ V/M | hv = 0}.

Applying this Lemma to the representation of 𝔰𝔩_{2} in V_{λ} we get

l_{λ}(f_{i}) = dim Ker ψ(f_{i}, ·) = dim M_{λ} − dim U_{λ} + dim Coker ψ(f_{i}, ·) =
= dim M_{λ} − dim U_{λ} + dim U_{λ}^{h_{i}}.

Notice that this formula remains valid for f_{i} = 0 if we assume that in this case h_{i} = 0. Joining this formula with (4.2) and applying the Lemma above we get the following result.

Theorem 4.13 ([T3]) Suppose that for any i

dim U_{λ}^{h_{i}} ≤ codim_{𝔤/𝔭} Lf_{i}.

Then generic global sections of the bundle L_{λ} have zeros.

All ingredients of the formula (4.13) can be easily computed in many particular cases. We will apply Theorem 4.13 for the proof of the following

Theorem 4.14 ([T3]) Let w ∈ S^{d}V^{∗}, resp. w ∈ Λ^{d}V^{∗}, be a generic form. Then V contains a k-dimensional isotropic subspace w.r.t. w if and only if

n ≥ \frac{\binom{d + k − 1}{d}}{k} + k, resp. n ≥ \frac{\binom{k}{d}}{k} + k,  (4.3)

with the following exceptions (a form is supposed to be generic). V contains a k-dimensional isotropic subspace if and only if – n ≥ 2k for w ∈ S^{2}V^{∗} or w ∈ Λ^{2}V^{∗}.

– k ≤ n − 2 for w ∈ Λ^{n−2}V^{∗}, n is even.

– k ≤ 4 for w ∈ Λ^{3}V^{∗}, n = 7.

It is interesting that the variety of 4-dimensional isotropic subspaces of a generic skewsymmetric 3-form in ℂ^{7} is a smooth 8-dimensional Fano variety. Moreover, this variety is a compactification of the unique symmetric space of the simple algebraic group G_{2}. Proof. We choose a basis {e_{1}, …, e_{n}} in V and identify GL_{n} with the group of non-singular matrices. Let T, B, B^{−} be the subgroups of diagonal, upperand lower-triangular matrices. We fix an integer k. Consider the parabolic subgroup

P = ( \begin{matrix}A & 0 \\ ∗ & B \end{matrix} ) ,

where B is a k × k-matrix. Then G/P is the Grassmanian Gr(k, V). Consider the vector bundle L = S^{d}S^{∗} (resp. L = Λ^{d}S^{∗}, but only in the case k ≥ d) on G/P, where S is the tautological bundle. Then L = L_{λ}, where λ is the highest weight of the GL_{n}-module S^{d}V^{∗} (resp. Λ^{d}V^{∗}). Let w ∈ S^{d}V^{∗} (resp. w ∈ Λ^{d}V^{∗}), let s_{w} be the corresponding global section. It is easy to see that (Z_{s_{w}})_{red} coincides with the variety of k-dimensional isotropic subspaces w.r.t. w. Notice that inequalities (4.3) are equivalent to the condition dim U_{λ} ≤ dim G/P. In the sequel we suppose that these inequalities hold.

We take

f_{i} = E_{1,n} + E_{2,n−1} + … + E_{i,n+1−i}, i = 0, …, r = min(k, n − k).

The unipotent radical 𝔭_{n}^{−} can be identified with matrices of shape k × (n − k) then the orbit Lf_{i} is identified with the variety of matrices of rank i, therefore

codim_{𝔤/𝔭} Lf_{i} = (k − i)(n − k − i).

We have

h_{i} = (E_{n,n} − E_{1,1}) + (E_{n−1,n−1} − E_{2,2}) + … + (E_{n+1−i,n+1−i} − E_{i,i}),

hence

dim U_{λ}^{h_{i}} = \binom{d + k − i − 1}{d} if V_{λ} = S^{d}V^{∗};

dim U_{λ}^{h_{i}} = \binom{k − i}{d} if V_{λ} = Λ^{d}V^{∗}.

Applying Theorem4.13 we get

Proposition 4.15 (a) Suppose that for any i = 0, …, r we have

\binom{d + k − i − 1}{d} ≤ (k − i)(n − k − i).

Then a generic form w ∈ S^{d}V^{∗} has a k-dimensional isotropic subspace.

(b) Suppose that for any i = 0, …, r we have

\binom{k − i}{d} ≤ (k − i)(n − k − i).

Then a generic form w ∈ Λ^{d}V^{∗} has a k-dimensional isotropic subspace.

It remains to clarify when the conditions of Proposition 4.15 follow from formulas (4.3).

A. Symmetric Case.

Clearly if i = k then the conditions of Proposition 4.15 are satisfied. Therefore it suffices to find out when the inequality

n ≥ \frac{\binom{d+k−i−1}{d}}{k − i} + k + i

follows from the inequality

n ≥ \frac{\binom{d+k−1}{d}}{k} + k

as i = 1, …, min(k −1, n−k). If d = 1 then Theorem 4.14 is obvious, if d = 2 it reduces to a well-known result about isotropic subspaces of quadratic form. So assume that d ≥ 3. We use a following Lemma that can be easily verified by induction:

Lemma 4.16 Let d ≥ 3, α ≥ 2. Then \frac{\binom{d+α−1}{d}}{α} ≥ \frac{\binom{d+α−2}{d}}{α−1} + 1.

It follows from Lemma 4.16 that

n ≥ \frac{\binom{d+k−1}{d}}{k} + k ≥ \frac{\binom{d+k−2}{d}}{k − 1} + k + 1

≥ \frac{\binom{d+k−3}{d}}{k − 2} + k + 2 ≥ … ≥ \frac{\binom{d+k−1−(k−1)}{d}}{k − (k − 1)} + k + k − 1.

QED.

B. Skew-symmetric Case.

Clearly if i > k − d then the conditions of Proposition 4.15 are satisfied. Therefore it suffices to check when the inequality

n ≥ \frac{\binom{k−i}{d}}{k − i} + k + i

follows from the inequality

n ≥ \frac{\binom{k}{d}}{k} + k

as i = 1, …, min(k −d, n−k). If d = 1 then Theorem 4.14 is obvious, if d = 2 it reduces to a well-known result about isotropic subspaces of a 2-form. So assume that d ≥ 3. We use a following Lemma that can be easily verified by induction:

Lemma 4.17 Let 3 ≤ d ≤ α − 2. Then \frac{\binom{α}{d}}{α} ≥ \frac{\binom{α−1}{d}}{α−1} + 1.

It follows from Lemma 4.17 that

n ≥ \frac{\binom{k}{d}}{k} + k ≥ \frac{\binom{k−1}{d}}{k − 1} + k + 1

≥ \frac{\binom{k−2}{d}}{k − 2} + k + 2 ≥ … ≥ \frac{\binom{k−(k−3−d)}{d}}{k − (k − 3 − d)} + k + (k − 3 − d).

It remains to consider 3 cases: i = k − d − 2, i = k − d − 1, i = k − d as i ≤ n − k.

Let i = k − d − 2, n ≥ 2k − d − 2. We want to deduce the inequality

n ≥ \frac{\binom{d+2}{d}}{d+2} + 2k − d − 2 = 2k − \frac{d−1}{2} − 2 from n ≥ \frac{\binom{k}{d}}{k} + k. It suffices to

check \binom{k}{d} ≥ k(k − \frac{d−1}{2} − 2). Since d ≥ 3 and k − d = i + 2 ≥ 3 we have \binom{k}{d} ≥ \binom{k}{3} ≥ k(k − 3) ≥ k(k − \frac{d−1}{2} − 2).

Let i = k − d − 1, n ≥ 2k − d − 1. We want to deduce the inequality

n ≥ \frac{\binom{d+1}{d}}{d+1} + 2k − d − 1 = 2k − d from n ≥ \frac{\binom{d+1}{d}}{d+1} + 2k − d − 1 = 2k − d.

It suffices to check \binom{k}{d} ≥ k(k − d). Since d ≥ 3 and k − d = i + 1 ≥ 2 we have \binom{k}{d} ≥ \binom{k}{3} ≥ k(k − 3) ≥ k(k − d) as k − d ≥ 3. If k − d = 2 then \binom{k}{d} = \frac{(d+2)(d+1)}{2} ≥ 2(d + 2) ≥ k(k − d).

Let i = k − d, n ≥ 2k − d. We need to deduce the inequality n ≥ \frac{\binom{d}{d}}{d} +

2k − d = 2k − d + \frac{1}{d} from n ≥ \frac{\binom{k}{d}}{k} + k. All exceptions are defined by the following system of equalities and inequalities:

n = 2k − d, \binom{k}{d} ≤ k(k − d), k − d ≥ 1.

There exist only two possibilities: either k = d + 1, n = d + 2 or d = 3, k = 5, n = 7.

In the first case we need to clarify whether a generic form w ∈ Λ^{n−2}V^{∗} has a (n − 1)-dimensional isotropic subspace or not. This holds iff a generic 2-form in V^{∗} has a non-zero kernel. It is well-known that this is true only for odd n.

To complete the proof it remains to find out whether a generic form w ∈ Λ^{3}(ℂ^{7})^{∗} has a 5-dimensional isotropic subspace. In suitable coordinates this isotropic subspace will coincide with the linear span ⟨e_{1}, …, e_{5}⟩. Therefore

w = ∑_{1≤i<j<k≤7} α_{ijk}x_{i} ∧ x_{j} ∧ x_{k}, where α_{ijk} = 0 as k ≤ 5. Consider the

one-parameter subgroup

H(t) = diag(t^{2}, t^{2}, t^{2}, t^{2}, t^{2}, t^{−5}, t^{−5})

in SL_{7}. Clearly, lim_{t→0} H(t)w = 0. Therefore w belongs to the Null-cone of the

action SL_{7} : Λ^{3}(ℂ^{7})^{∗}. It follows that any non-constant homogeneous invariant of this action vanishes at w. But it is known that this action has non-constant invariants (see [PV]), for example, the discriminant is well-defined in this case. Therefore generic forms do not admit 5-dimensional isotropic subspaces.

Moreover, it can be shown that a 3-form in the 7-dimensional vector space admits a 5-dimensional isotropic subspace if and only if its discriminant is equal to zero. ⊓ ⊔

4.4 Moore–Penrose Inverse and Applications

The nice notion of a generalized inverse of an arbitrary matrix (possibly singular or even non-square) has been discovered independently by Moore [Mo] and Penrose [Pe]. The following definition belongs to Penrose (Moore’s definition is different but equivalent):

Definition 4.18 A matrix A^{+} is called a MP-inverse of a matrix A if

AA^{+}A = A, A^{+}AA^{+} = A^{+},

and AA^{+}, A^{+}A are Hermitian matrices.

It is quite surprising but a MP-inverse always exists and is unique. Since the definition is symmetric with respect to A and A^{+} it follows that (A^{+})^{+} = A. If A is a non-singular square matrix then A^{+} coincides with an ordinary inverse matrix A^{−1}. The theory of MP-inverses and their numerous modifications becomes now a separate subfield of Linear Algebra [CM] with various applications. Here we show that this notion quite naturally arises in the theory of shortly graded simple Lie algebras and give applications. To explain this connection let us first give another definition of a MP-inverse.

Let A ∈ Mat_{n,m}(ℂ). Then it is easy to see that a matrix A^{+} ∈ Mat_{m,n}(ℂ) is a MP-inverse of A if and only if there exist Hermitian matrices

B_{1} ∈ Mat_{n,n}(ℂ) and B_{2} ∈ Mat_{m,m}(ℂ)

such that the following matrices form an 𝔰𝔩_{2}-triple in 𝔰𝔩_{n+m}(ℂ):

E = ( \begin{matrix}0 & A \\ 0 & 0 \end{matrix} ) , H = ( \begin{matrix}B_{1} & 0 \\ 0 & B_{2} \end{matrix} ) , F = ( \begin{matrix}0 & 0 \\ A^{+} & 0 \end{matrix} ) .

By an 𝔰𝔩_{2}-triple ⟨e, h, f⟩ in a Lie algebra 𝔤 we mean a collection of (possibly zero) vectors such that

[e, f] = h, [h, e] = 2e, [h, f] = −2f.

In other words, an 𝔰𝔩_{2}-triple is a homomorphic image of canonical generators of 𝔰𝔩_{2} with respect to some homomorphism of Lie algebras 𝔰𝔩_{2} → 𝔤.

This definition admits an immediate generalization. Suppose that 𝔤 is a simple complex Lie algebra, G is a corresponding simple simply-connected Lie group. Suppose further that P is a parabolic subgroup of G with abelian unipotent radical (with aura). Then 𝔤 admits a short grading

𝔤 = 𝔤_{−1} ⊕ 𝔤_{0} ⊕ 𝔤_{1}

with only three nonzero parts. Here 𝔭 = 𝔤_{0} ⊕ 𝔤_{1} is a Lie algebra of P and exp 𝔤_{1} is the abelian unipotent radical of P. Let 𝔨_{0} be a compact real form of 𝔤_{0}. In this section we shall permanently consider compact real forms of reductive subalgebras of simple Lie algebras. These subalgebras will always be Lie algebras of algebraic reductive subgroups of a corresponding simple complex algebraic group. Their compact real forms will always be understood as Lie algebras of compact real forms of corresponding algebraic groups. For example, a Lie algebra of an algebraic torus has a unique compact real form.

Suppose now that e ∈ 𝔤_{1}. It is well-known that there exists a homogeneous 𝔰𝔩_{2}-triple ⟨e, h, f⟩ such that h ∈ 𝔤_{0} and f ∈ 𝔤_{−1}.

Definition 4.19 An element f ∈ 𝔤_{−1} is called a MP-inverse of e ∈ 𝔤_{1} if there exists a homogeneous 𝔰𝔩_{2}-triple ⟨e, h, f⟩ with h ∈ i𝔨_{0}.

MP-inverses of elements f ∈ 𝔤_{−1} are defined in the same way. It is clear that if f is a MP-inverse of e then e is a MP-inverse of f.

Example 4.20 Suppose that G = SL_{n+m} and P ⊂ G is a maximal parabolic subgroup of block triangular matrices of the form

( \begin{matrix}B_{1} & A \\ 0 & B_{2} \end{matrix} ) , where B_{1} ∈ Mat_{n,n}, A ∈ Mat_{n,m}, B_{2} ∈ Mat_{m,m} .

The graded components of the correspondent grading consist of matrices of the following form:

𝔤_{−1} = ( \begin{matrix}0 & 0 \\ A′ & 0 \end{matrix} ) , 𝔤_{0} = ( \begin{matrix}B_{1} & 0 \\ 0 & B_{2} \end{matrix} ) , 𝔤_{1} = ( \begin{matrix}0 & A \\ 0 & 0 \end{matrix} ) ,

where A′ ∈ Mat_{m,n}, B_{1} ∈ Mat_{n,n}, B_{2} ∈ Mat_{m,m}, and A ∈ Mat_{n,m}. One can take 𝔨_{0} to be the real Lie algebra of block diagonal skew-Hermitian matrices with zero trace. Then i𝔨_{0} is a vector space of block diagonal Hermitian matrices with zero trace. Therefore in this case we return to a previous definition of a Moore–Penrose inverse.

Theorem 4.21 ([T4]) For any e ∈ 𝔤_{1} there exists a unique Moore–Penrose inverse f ∈ 𝔤_{−1}.

The proof is similar to the proof of Theorem 4.23 below.

It obviously follows that for any non-zero f ∈ 𝔤_{−1} there exists a unique MP-inverse e ∈ 𝔤_{1}. So taking a MP-inverse is a well-defined involutive operation. In general, it is not equivariant with respect to a Levi subgroup L ⊂ P with Lie algebra 𝔤_{0}, but only with respect to its maximal compact subgroup K_{0} ⊂ L.

Let us give an intrinsic description of the Moore-Penrose inverse in all cases arising arising from short gradings of classical simple Lie algebras. Exceptional cases may be found in [T4].

Linear Maps.

This is, of course, the classical Moore–Penrose inverse. Let us recall its intrinsic description. Suppose that ℂ^{n} and ℂ^{m} are vector spaces equipped with standard Hermitian scalar products. For any linear map F : ℂ^{n} → ℂ^{m} its Moore-Penrose inverse is a linear map F^{+} : ℂ^{m} → ℂ^{n} defined as follows. Let Ker F ⊂ ℂ^{n} and Im F ⊂ ℂ^{m} be a kernel and an image of F. Let Ker^{⊥} F ⊂ ℂ^{n} and Im^{⊥} F ⊂ ℂ^{m} be their orthogonal complements with respect to the Hermitian scalar products. Then F defines via restriction a bijective linear map F̃ : Ker^{⊥} F → Im F. Then F^{+} : ℂ^{m} → ℂ^{n} is a unique linear map such that F^{+}|_{Im^{⊥} F} = 0 and F^{+}|_{Im F} = F̃^{−1}. This MP-inverse corresponds to short gradings of 𝔰𝔩_{n+m}.

Symmetric and Skew-symmetric Bilinear Forms.

Let V = ℂ^{n} be a vector space equipped with a standard Hermitian scalar product. For any symmetric (resp. skew-symmetric) bilinear form ω on V its Moore-Penrose inverse is a symmetric (resp. skew-symmetric) bilinear form ω^{+} on V^{∗} defined as follows. Let Ker ω ⊂ V be the kernel of ω. Then ω induces a non-degenerate bilinear form ω̃ on V/ Ker ω. Let Ann(Ker ω) ⊂ V^{∗} be an annihilator of Ker ω. Then Ann(Ker ω) is canonically isomorphic to the dual of V/ Ker ω. Therefore the form ω̃^{−1} on Ann(Ker ω) is well-defined. The form ω^{+} is defined as a unique form such that its restriction on on Ann(Ker ω) coincides with ω̃^{−1} and its kernel is Ann(Ker ω)^{⊥}, the orthogonal complement with respect to a standard Hermitian scalar product on V^{∗}. This MP-inverse corresponds to the short grading of 𝔰𝔭_{2n+2} (resp. 𝔰𝔬_{2n+2}).

Vectors in a Vector Space With the Scalar Product.

Suppose that V = ℂ^{n} is a vector space with the standard bilinear scalar product (·, ·). For any vector v ∈ V its Moore-Penrose inverse v^{∨} is again a vector in V defined as follows:

v^{∨} = \{ \begin{matrix}\frac{2v}{(v, v)}, & if (v, v) ≠ 0 \\ \frac{\overline{v}}{(\overline{v}, v)}, & if (v, v) = 0, v ≠ 0 \\ 0, & if v = 0. \end{matrix}

This MP-inverse corresponds to the short grading of 𝔰𝔬_{n+2}.

It is quite natural to ask whether it is possible to extend the notion of the Moore–Penrose inverse from parabolic subgroups with aura to arbitrary parabolic subgroups. It is also interesting to consider the “non-graded” situation. Let us start with it. Suppose G is a simple connected simply-connected Lie group with Lie algebra 𝔤. We fix a compact real form 𝔨 ⊂ 𝔤.

Definition 4.22 A nilpotent orbit O ⊂ 𝔤 is called a Moore–Penrose orbit if for any e ∈ O there exists an 𝔰𝔩_{2}-triple ⟨e, h, f⟩ such that h ∈ i𝔨.

It turns out that it is quite easy to find all Moore-Penrose orbits. Recall that the height ht(O) of a nilpotent orbit O = Ad(G)e is equal to the maximal integer k such that ad(e)^{k} ≠ 0. Clearly ht(O) ≥ 2.

Theorem 4.23 O is a Moore–Penrose orbit if and only if ht(O) = 2. In this case for any e ∈ O there exists a unique 𝔰𝔩_{2}-triple ⟨e, h, f⟩ such that h ∈ i𝔨.

Proof. Let x → \overline{x} denotes a complex conjugation in 𝔤 with respect to the compact form 𝔨. Therefore x = \overline{x} iff x ∈ 𝔨 and x = −\overline{x} iff x ∈ i𝔨. Let B(x, y) = Tr ad(x) ad(y) be the Killing form of 𝔤. Finally, let H(x, y) = −B(x, \overline{y}) be a positive-definite Hermitian form on 𝔤.

Lemma 4.24 We fix a nilpotent element e ∈ 𝔤. Suppose that ⟨e, h, f⟩ is an 𝔰𝔩_{2}-triple in 𝔤 such that h ∈ i𝔨. Then for any other 𝔰𝔩_{2}-triple ⟨e, h′, f′⟩ we have H(h, h) < H(h′, h′). In particular, if there exists an 𝔰𝔩_{2}-triple ⟨e, h, f⟩ with h ∈ i𝔨 then the 𝔰𝔩_{2}-triple with this property is unique.

Proof. Recall that if ⟨e, h, f⟩ is an 𝔰𝔩_{2}-triple then h is called a characteristic of e. Consider the subset H ⊂ 𝔤 consisting of all possible characteristics of e. It is well-known that H is an affine subspace in 𝔤 such that the corresponding linear subspace is precisely the unipotent radical 𝔷_{𝔤}^{u}(e) of the centralizer 𝔷_{𝔤}(e) in 𝔤 of the element e. Since H(h′, h′) is a strongly convex function on H, there exists a unique element h_{0} ∈ H such that H(h_{0}, h_{0}) < H(h′, h′) for any h′ ∈ H, h′ ≠ h_{0}. We need to show that h_{0} = h. It is clear that an element h_{0} ∈ H minimizes H(h, h) on H iff H(h_{0}, 𝔷_{𝔤}^{u}(e)) = 0 iff B(\overline{h}_{0}, 𝔷_{𝔤}^{u}(e)) = 0. If h ∈ H ∩ i𝔨_{0} then \overline{h} = −h and we have

B(\overline{h}, 𝔷_{𝔤}^{u}(e)) = −B(h, 𝔷_{𝔤}^{u}(e)) = −B([e, f], 𝔷_{𝔤}^{u}(e)) = B(f, [e, 𝔷_{𝔤}^{u}(e)]) = 0.

Therefore h = h_{0}. ⊓ ⊔ Suppose that ⟨e, h, f⟩ is an 𝔰𝔩_{2}-triple in 𝔤. Consider the grading 𝔤 = ⊕_{k} 𝔤_{k}

such that x ∈ 𝔤_{k} iff [h, x] = kx. Let 𝔫_{+} = ⊕_{k>0} 𝔤_{k}, 𝔫_{−} = ⊕_{k<0} 𝔤_{k}. It is well

known that 𝔷_{𝔤}^{u}(e) ⊂ 𝔫_{+}.

Lemma 4.25 Suppose that 𝔷_{𝔤}^{u}(e) = 𝔫_{+}. Then O = Ad(G)e is a Moore– Penrose orbit.

Proof. We need to prove that for any element e′ ∈ O there exists an 𝔰𝔩_{2}- triple ⟨e′, h′, f′⟩ such that h′ ∈ i𝔨, where 𝔨 is a fixed compact real form of 𝔤. Clearly it is sufficient to prove that for an arbitrary compact real form 𝔨 there exists an 𝔰𝔩_{2}-triple ⟨e, h, f⟩ with h ∈ i𝔨. According to the proof of the previous Lemma we should choose h to be a unique characteristic such that B(\overline{h}, 𝔷_{𝔤}^{u}(e)) = 0, where x → \overline{x} denotes a complex conjugation in 𝔤 with respect to the compact form 𝔨. It remains to prove that h ∈ i𝔨. Since B is a non-degenerate ad-invariant scalar product on 𝔤 and 𝔷_{𝔤}^{u}(e) = 𝔫_{+} it follows that \overline{h} ∈ 𝔭, where 𝔭 = 𝔤_{0} ⊕𝔫_{+}. Let 𝔩 be some “standard” compact real form of 𝔤 such that h ∈ i𝔩 and 𝔫̃_{±} = 𝔫_{∓}, where x → x̃ denotes a complex conjugation in 𝔤 with respect to the compact form 𝔩. Let P ⊂ G be a parabolic subgroup of G with the Lie algebra 𝔭, let H ⊂ P be its Levi subgroup with the Lie algebra 𝔤_{0}. Since all compact real forms of a semisimple complex Lie algebra are conjugated by elements of any fixed Borel subgroup it follows that there exists g ∈ P such that Ad(g)𝔨 = 𝔩. Therefore

{ \atop Ad(g)h} = Ad(g)\overline{h} ⊂ Ad(g)(𝔭) = 𝔭.

We can express g as a product uz, where u ∈ exp(𝔫_{+}), Ad(z)h = h. Then { \atop Ad(g)h} = { \atop Ad(u)h}. If u is not the identity element of G then Ad(u)h = h + ξ, where ξ ∈ 𝔫_{+} and ξ ≠ 0. Therefore { \atop Ad(u)h} = −h + ξ̃. But ξ̃ ∈ 𝔫_{−} and hence { \atop Ad(u)h} ∉ 𝔭, contradiction. Therefore u is trivial and since Ad(z)h = h we finally get

\overline{h} = h̃ = −h.⊓ ⊔
⊓ ⊔

Now we shall try to reverse this argument.

Lemma 4.26 Suppose that O = Ad(G)e is a Moore–Penrose orbit. Then 𝔷_{𝔤}^{u}(e) = 𝔫_{+}.

Proof. We choose a standard compact real form 𝔩 as in the proof of the previous Lemma. Clearly, 𝔷_{𝔤}^{u}(e) is a graded subalgebra of 𝔫_{+} = ⊕_{k>0} 𝔤_{k}. Suppose, on

the contrary, that 𝔷_{𝔤}^{u}(e) ≠ 𝔫_{+}. Let ξ ∈ 𝔤_{p}, p > 0, be a homogeneous element that does not belong to 𝔷_{𝔤}^{u}(e). Let u = exp(ξ). Let e′ = Ad(u)e. We claim that all characteristics of e′ do not belong to i𝔩. Indeed, all characteristics of e′ have a form Ad(u)h+Ad(u)x, where x ∈ 𝔷_{𝔤}^{u}(e). Suppose that for some x we have Ad(u)h+Ad(u)x ∈ i𝔩. Since h ∈ i𝔩, 𝔫̃_{±} = 𝔫_{∓}, and Ad(u)(h+x)−h ∈ 𝔫_{+} it follows that Ad(u)(h+x) = h. In 𝔫_{+} modulo ⊕_{k>p} 𝔤_{k} we obtain the equation

[ξ, h] + x = 0, but [h, ξ] = pξ and therefore ξ ∈ 𝔷_{𝔤}^{u}(e). Contradiction. ⊓ ⊔

Now we can finish the proof of Theorem 4.23. Combining previous lemmas we see that O is a Moore–Penrose orbit if and only if 𝔷_{𝔤}^{u}(e) = 𝔫_{+}. It follows from the 𝔰𝔩_{2}-theory that dim 𝔷_{𝔤}^{u}(e) = dim 𝔤_{1} + dim 𝔤_{2}. Therefore 𝔷_{𝔤}^{u}(e) = 𝔫_{+} if and only if 𝔤_{p} = 0 for p > 2. Clearly, this is precisely equivalent to ht(O) = 2. In this case for any e ∈ O there exists a unique 𝔰𝔩_{2}-triple ⟨e, h, f⟩ such that h ∈ i𝔨 by Lemma 4.24. ⊓ ⊔

Now let us turn to the graded situation. Suppose that 𝔤 is a ℤ-graded simple Lie algebra, 𝔤 = ⊕_{k∈ℤ} 𝔤_{k}. Let P ⊂ G be a parabolic subgroup with Lie

algebra 𝔭 = ⊕_{k≥0} 𝔤_{k}. Let L ⊂ P be a Levi subgroup with Lie algebra 𝔤_{0}. We

choose a compact real form 𝔨_{0} of 𝔤_{0}. Suppose now that e ∈ 𝔤_{k}. It is wellknown that there exists a homogeneous 𝔰𝔩_{2}-triple ⟨e, h, f⟩ with h ∈ 𝔤_{0} and f ∈ 𝔤_{−k}.

Definition 4.27 Take any k > 0 and any L-orbit O ⊂ 𝔤_{k}. Then O is called a Moore-Penrose orbit if for any e ∈ O there exists a homogeneous 𝔰𝔩_{2}-triple ⟨e, h, f⟩ such that h ∈ i𝔨_{0}. In this case f is called a MP-inverse of e. A grading is called a Moore–Penrose grading in degree k > 0 if all L-orbits in 𝔤_{k} are Moore-Penrose. A grading is called a Moore–Penrose grading if it is a Moore–Penrose grading in any positive degree. A parabolic subgroup P ⊂ G is called a Moore–Penrose parabolic subgroup if there exists a Moore–Penrose grading 𝔤 = ⊕_{k∈ℤ} 𝔤_{k} such that 𝔭 = ⊕_{k≥0} 𝔤_{k} is a Lie algebra of P.

One should be careful comparing graded and non-graded situation: if O ⊂ 𝔤_{k} is a Moore–Penrose L-orbit then Ad(G)O ⊂ 𝔤 is not necessarily a Moore– Penrose G-orbit. Let us give a criterion for an L-orbit to be Moore–Penrose. Suppose O = Ad(L)e ⊂ 𝔤_{k}. Take any homogeneous 𝔰𝔩_{2}-triple ⟨e, h, f⟩. Then h defines a grading 𝔤_{0} = ⊕_{n∈ℤ} 𝔤_{0}^{n}, such that ad(h)|_{𝔤_{0}^{n}} = n · Id.

Theorem 4.28 ([T4]) O is a Moore–Penrose orbit if and only if ad(e)g_{0}^{n} = 0 for any n > 0. In this case for any e′ ∈ O there exists a unique homogeneous 𝔰𝔩_{2}-triple ⟨e′, h′, f′⟩ such that h′ ∈ i𝔨_{0}.

The proof is similar to the proof of Theorem 4.23.

Example 4.29 Suppose that G is a simple group of type G_{2}. We fix a root decomposition. There are two simple roots α_{1} and α_{2} such that α_{1} is short and α_{2} is long. There are 3 proper parabolic subgroups: Borel subgroup B and two maximal parabolic subgroups P_{1} and P_{2} such that a root vector of α_{i} belongs to a Levi subgroup of P_{i}. Then the following is an easy application of Theorem 4.28. B is a Moore–Penrose parabolic subgroup (actually Borel subgroups in all simple groups are Moore–Penrose parabolic subgroups with respect to any grading). P_{1} is not Moore–Penrose, but it is a Moore– Penrose parabolic subgroup in degree 2 (with respect to the natural grading of height 2). P_{2} is a Moore–Penrose parabolic subgroup.

Example 4.30 Suppose G = SL_{n}. We fix positive integers d_{1}, …, d_{k} such that n = d_{1} + … + d_{k}. We consider the parabolic subgroup P (d_{1}, …, d_{k}) ⊂ SL_{n} that consists of all upper-triangular block matrices with sizes of blocks equal to d_{1}, …, d_{k}. We take a standard grading. Then 𝔤_{1} is identified with the linear space of all tuples of linear maps {f_{1}, …, f_{k}},

ℂ^{d_{1}} ⟵^{f_{1}} ℂ^{d_{2}} ⟵^{f_{2}} … ⟵^{f_{k−1}} ℂ^{d_{k}},

𝔤_{−1} is identified with the linear space of all tuples of linear maps {g_{1}, …, g_{k}},

ℂ^{d_{1}} ⟶^{g_{1}} ℂ^{d_{2}} ⟶^{g_{2}} … ⟶^{g_{k−1}} ℂ^{d_{k}},

and Levi subgroup L(d_{1}, …, d_{k}) is just the group

(A_{1}, …, A_{k}) ∈ GL_{d_{1}} × … × GL_{d_{k}} such that det(A_{1}) · . . . · det(A_{k}) = 1.

It acts on these spaces of linear maps in an obvious way. The most important among L-orbits are varieties of complexes. To define them, let us fix in addition non-negative integers m_{1}, …, m_{k−1} such that m_{i−1} + m_{i} ≤ d_{i} (we set m_{0} = m_{k} = 0), and consider the subvariety of all tuples {f_{1} …, f_{k−1}} as above such that rk f_{i} = m_{i} and f_{i−1}∘f_{i} = 0 for any i. These tuples form a single L-orbit O called a variety of complexes. For any tuple {f_{1}, …, f_{k−1}} ∈ O consider the tuple {f_{1}^{+}, …, f_{k−1}^{+}} ∈ 𝔤_{−1}, where f_{i}^{+} is a classical “matrix” Moore–Penrose inverse of f_{i}. It is easy to see that this new tuple is again a complex, moreover, this complex is a Moore–Penrose inverse (in our latest meaning of this word) of an original complex. In particular, orbits of complexes are Moore–Penrose orbits.

From the first glance only few parabolic subgroups are Moore–Penrose. But this is scarcely true. For example, we have the following Theorem:

Theorem 4.31 ([T4]) Any parabolic subgroup in SL_{n} is Moore–Penrose.

To explain our interest in Moore–Penrose parabolic subgroups let us recall Conjecture 4.11 from the previous section. Suppose once again that G is a simple connected simply-connected Lie group, P is its parabolic subgroup, 𝔭 ⊂ 𝔤 are their Lie algebras. We take any irreducible G-module V. There exists a unique proper P -submodule M_{V} of V. We have the inclusion i : M_{V} → V, the projection π : V → V/M_{V} and the map R_{V} : 𝔤 → End(V) defining the representation. Therefore we have a linear map R̃_{V} : 𝔤 → Hom(M_{V} , V/M_{V}), namely R̃_{V} (x) = π ∘ R_{V} (x) ∘ i. Clearly 𝔭 ⊂ Ker R̃_{V}. Finally, we have a linear map Ψ_{V} : 𝔤/𝔭 → Hom(M_{V} , V/M_{V}). The Conjecture 4.11 states that there exists an algebraic stratification 𝔤/𝔭 = ⊔_{i=1}^{n} X_{i} such that for any V the function rk Ψ_{V} is constant along each X_{i}.

The following theorem shows the connection of this problem with the Moore–Penrose inverse.

Theorem 4.32 ([T4]) Suppose that a grading 𝔤 = ⊕_{k∈ℤ} 𝔤_{k} is a Moore–

Penrose grading in all positive degrees except at most one. Then the Conjecture is true for the corresponding parabolic subgroup P.

Proof. Suppose that G is a connected reductive group with a Lie algebra 𝔤. For any elements x_{1}, …, x_{r} ∈ 𝔤 let ⟨x_{1}, …, x_{r}⟩_{alg} denote the minimal algebraic Lie subalgebra of 𝔤 that contains x_{1}, …, x_{r} (algebraic subalgebras are the Lie algebras of algebraic subgroups). By a theorem of Richardson [Ri1] ⟨x_{1}, …, x_{r}⟩_{alg} is reductive if and only if an orbit of the r-tuple (x_{1}, …, x_{r}) in 𝔤^{r} is closed with respect to the diagonal action of G. Suppose now that h_{1}, …, h_{r} are semi-simple elements of 𝔤. Consider the closed subvariety Ô = (Ad(G)h_{1}, …, Ad(G)h_{r}) ⊂ 𝔤^{r}. For any closed G-orbit O ⊂ Ô let us denote by G(O) the conjugacy class of the reductive subalgebra ⟨x_{1}, …, x_{r}⟩_{alg} for (x_{1}, …, x_{r}) ∈ O.

Lemma 4.33 There are only finitely many conjugacy classes G(O).

Proof. We shall use induction on dim 𝔤. Suppose that the claim of the Lemma is true for all reductive groups H with dim H < dim G. Let 𝔷 ⊂ 𝔤 be the center of 𝔤, 𝔤′ ⊂ 𝔤 be its derived algebra. Consider two canonical homomorphisms

𝔤 ⟶^{π} 𝔤′ and 𝔤 ⟶^{π′} 𝔷.

We take any closed G-orbit O ⊂ Ô. Let (x_{1}, …, x_{r}) ∈ O, y_{i} = π(x_{i}) for i = 1, …, r. Then ⟨y_{1}, …, y_{r}⟩_{alg} = π(⟨x_{1}, …, x_{r}⟩_{alg}) and, therefore, is reductive. Let us consider two cases.

Suppose first, that ⟨y_{1}, …, y_{r}⟩_{alg} = 𝔤′. Then 𝔤′ is a derived algebra of ⟨x_{1}, …, x_{r}⟩_{alg} and, therefore, ⟨x_{1}, …, x_{r}⟩_{alg} = ⟨π′(h_{1}), …, π′(h_{r})⟩ ⊕ 𝔤′. In this case we get one conjugacy class.

Suppose now, that ⟨y_{1}, …, y_{r}⟩_{alg} ≠ 𝔤′. Then ⟨y_{1}, …, y_{r}⟩_{alg} is contained in some maximal reductive Lie subalgebra of 𝔤′. It is well-known (and not difficult to prove) that in a semisimple Lie algebra there are only finitely many conjugacy classes of maximal reductive subalgebras. Let 𝔥′ be one of them, 𝔥 = 𝔷⊕𝔥′ ⊂ 𝔤. Let H be a corresponding reductive subgroup of G. It is sufficient to prove that for any closed G-orbit O of Ô that meets 𝔥^{r} there are only finitely many possibilities for G(O). It easily follows from Richardson’s Lemma [Ri] that for any i the intersection Ad(G)h_{i} ∩ 𝔥 is a union of finitely many closed H-orbits, say Ad(H)h_{i}^{1}, …, Ad(H)h_{i}^{s_{i}}. It remains to prove that if for some r-tuple (x_{1}, …, x_{r}) ∈ Ad(H)h_{1}^{k_{1}} × … Ad(H)h_{r}^{k_{r}} the corresponding subalgebra ⟨x_{1}, …, x_{r}⟩_{alg} is reductive then there are only finitely many possibilities for its conjugacy class. But this is precisely the claim of Lemma for the group H, which is true by the induction hypothesis. ⊓ ⊔

Suppose that 𝔨 is a compact real form of 𝔤.

Lemma 4.34 If r-tuple (x_{1}, …, x_{r}) belongs to (i𝔨)^{r}, then its G-orbit is closed in 𝔤^{r}.

Proof. Indeed, let B be a non-degenerate ad-invariant scalar product on 𝔤, which is negative-definite on 𝔨. Let H(x) = −B(\overline{x}, x) be a positive-definite 𝔨-invariant Hermitian quadratic form on 𝔤, where the complex conjugation is taken with respect to 𝔨. Let H^{r} be a corresponding Hermitian quadratic form on 𝔤^{r}. More precisely, H^{r}(x_{1}, …, x_{r}) = H(x_{1}) + … + H(x_{r}). By a Kempf– Ness criterion [PV] in order to prove that the G-orbit of (x_{1}, …, x_{r}) is closed it is sufficient to prove that the real function H^{r}(·) has a critical point on this orbit. Let us show that (x_{1}, …, x_{r}) is this critical point. Indeed, for any g ∈ 𝔤

−B(\overline{x}_{1}, [g, x_{1}]) − … − B(\overline{x}_{r}, [g, x_{r}]) = B(x_{1}, [g, x_{1}]) + … + B(x_{r}, [g, x_{r}]) = 0.

⊓ ⊔

Now let G be a simple simply-connected Lie group, let 𝔤 be its Lie algebra with a ℤ-grading 𝔤 = ⊕_{k∈ℤ} 𝔤_{k}. Let r be a maximal integer such that 𝔤_{r} ≠ 0.

We denote the non-positive part of the grading ⊕_{k≤0} 𝔤_{k} by 𝔭. Let P ⊂ G be a

parabolic subgroup with the Lie algebra 𝔭. We shall identify 𝔤/𝔭 with ⊕_{k>0} 𝔤_{k}.

Let L ⊂ G be a connected reductive subgroup with Lie algebra 𝔤_{0}. Let V be an irreducible G-module. If we choose a Cartan subalgebra 𝔱 ⊂ 𝔤_{0} then the grading of 𝔤 originates from some ℤ-grading on 𝔱^{∗}. Therefore, there exists a ℤ-grading V = ⊕_{k∈ℤ} V_{k} such that 𝔤_{i}V_{j} ⊂ V_{i+j}. Let R be a maximal integer

such that V_{R} ≠ 0. It is easy to see that M_{V} = ⊕_{k<R} V_{k} (notice that V_{R} is an

irreducible L-module).

Now we can prove the Theorem. It is sufficient to prove that there exists a finite set of points {x_{1}, …, x_{N}} ⊂ 𝔤/𝔭 such that for any x ∈ 𝔤/𝔭 and for any V we have rk Ψ_{V} (x) = rk Ψ_{V} (x_{i}) for some i. Recall that L has finitely many orbits on each 𝔤_{k}, see Theorem 2.10. We pick some L-orbit O_{i} in each 𝔤_{i}. Then it is sufficient to find a finite set of points as above only for points x ∈ 𝔤/𝔭 of a form x = x_{1} + … + x_{r}, where x_{i} ∈ O_{i}. For any orbit O_{i} let H_{i} denote the set of all possible homogeneous characteristics of all elements from O_{i}. Clearly H_{i} is a closed Ad(L)-orbit. Let Ô = H_{1} × … × H_{r} ⊂ 𝔤_{0}^{r}. Then by the first Lemma the set of conjugacy classes of subgroups G(O) for closed L-orbits O in Ô is finite. Let us show that for any r-tuple (x_{1}, …, x_{r}) ∈ O_{1} × … × O_{r} there exists an r-tuple (h_{1}, …, h_{r}) ∈ Ô such that h_{i} is a homogeneous characteristic of x_{i} and an L-orbit Ad(L)(h_{1}, …, h_{r}) is closed. Indeed, after simultaneous conjugation of elements x_{i} by some element g ∈ L we may suppose that any x_{i} has a homogeneous characteristic h_{i} ∈ i𝔨_{0} (in all degrees except at most one no conjugation is needed because of Moore– Penrose property, for one degree this is obvious). Then by the second Lemma an orbit Ad(L)(h_{1}, …, h_{r}) is closed. Since all functions rk Ψ_{V} are L-invariant, we may restrict ourselves to the points x = ∑_{i} x_{i} ∈ 𝔤/𝔭 such that x_{i} ∈ O_{i}, any x_{i} has a homogeneous characteristic h_{i} ∈ i𝔨_{0}, and a conjugacy class of ⟨h_{1}, …, h_{r}⟩_{alg} is fixed. We claim that any function rk Ψ_{V} is constant along the set of these points. Moreover, we shall prove that

rk Ψ_{V} (x) = dim V_{R} − dim V_{R}^{⟨h_{1},...,h_{r}⟩_{alg}}.  (4.4)

Indeed,

rk Ψ_{V} (x) = dim ∑_{i} Im (ad(x_{i})|_{V_{R−i}}) .

Clearly V_{R} is ad(h_{i})-invariant and is killed by ad(e_{i}), therefore from the 𝔰𝔩_{2}- theory we get that V_{R} = ⊕_{k≥0} V_{R}^{k}, where ad(h_{i})|_{V_{R}^{k}} = k · Id. Moreover,

Im (ad(x_{i})|_{V_{R−i}}) = ⊕_{k>0} V_{R}^{k}.

Let H be a contravariant Hermitian form on V_{R} with respect to the compact form 𝔨_{0} of 𝔤_{0}. Since h_{i} ∈ i𝔨_{0} and H is a contravariant form we get that ⊕_{k>0} V_{R}^{k} = (V_{R}^{0})^{⊥}. Therefore,

∑_{i} Im (ad(x_{i})|_{V_{R−i}}) = (∩_{i}V_{R}^{h_{i}})^{⊥} = (V_{R}^{⟨h_{1},...,h_{r}⟩_{alg}})^{⊥} .

The formula (4.4) follows. ⊓ ⊔

For example, combining Theorem 4.32 and Theorem 4.31 we get the following corollary:

Corollary 4.35 The conjecture is true for any parabolic subgroup in SL_{n}.

Though the conditions of Theorem 4.32 are not always satisfied, it seems that one can prove the conjecture for any simple group in this direction.

5. Fulton–Hansen Theorem and Applications

Preliminaries

Fulton–Hansen connectedness Theorem has numerous applications to the projective geometry including famous Zak Theorem on tangencies. Aside from dual varieties, projective geometry also studies secant and tangential varieties, joins, etc. In this chapter we review some basic results related to them.

5.1 Fulton–Hansen Connectedness Theorem

Theorem 5.1 ([FH]) Let f : X → ℙ^{N} be a finite morphism from an irreducible complete variety X to the projective space ℙ^{N}. Then for any projective subspace L_{0} ⊂ ℙ^{N} such that dim X > codim_{ℙ^{N}} L_{0} the inverse image f^{−1}(L_{0}) is connected.

Sketch of the proof. Let k = dim L. Let U ⊂ Gr(k, ℙ^{N}) be the open set of all subspaces that meet f(X) properly, and let U_{0} ⊂ Gr(k, ℙ^{N}) be the set of all subspaces L such that f^{−1}(L) is irreducible. Then U_{0} is non-empty by Bertini’s theorem (see [Ha1] or [Wei]). It follows from Zariski’s principle of degeneration (see [Fu1] or [Za]) that for all L ∈ U the preimage f^{−1}(L) is connected, non-empty, of codimension dim X + k − N. For each L ∈ U, there is a positive cycle f^{∗}(L) on X whose support is f^{−1}(L). This determines a morphism from U to the Chow variety Z of cycles on X. Let

Γ ⊂ Gr(k, ℙ^{N}) × Z

be the closure of the graph of this morphism, π : Γ → Gr(k, ℙ^{N}) the projection. Since Z is complete, π is proper as well as birational. Zariski’s principle of degeneration implies that for any γ ∈ Γ the corresponding cycle C_{γ} is connected (i.e. its support |C_{γ}| is connected). Zariski’s main theorem implies that π^{−1}(L_{0}) is connected. It follows that the union of the |C_{γ}| for γ ∈ π^{−1}(L_{0}) is connected. It remains to prove that

f^{−1}(L_{0}) = ⋃_{γ∈π^{−1}(L_{0})} |C_{γ}|.

The inclusion ⊃ follows from the continuity of limit cycles. So it remains to prove the inclusion ⊂. Let x ∈ f^{−1}(L_{0}) and let z = f(x). Consider the subvariety G of Gr(k, ℙ^{N}) consisting of all subspaces passing through z (therefore G is isomorphic to Gr(k − 1, ℙ^{N−1}). An easy dimension count shows that G meets U. Let Γ′ ⊂ Γ be an irreducible subvariety that maps birationally to G. For each L ⊂ G ∩ U, the cycle f^{∗}(L) contains x, therefore, for any γ ∈ Γ′ the cycle C_{γ} contains x. In particular, there exists the limit cycle C_{γ} for γ ∈ Γ′ ∩ π^{−1}(L_{0}) that contains x. ⊓ ⊔

Now we shall derive a number of corollaries.

Corollary 5.2 ([FH]) Let f : X → ℙ^{N} be a morphism from an irreducible complete variety X. Then for any projective subspace L ⊂ ℙ^{N} such that dim f(X) > codim_{ℙ^{N}} L the inverse image f^{−1}(L) is connected.

Proof. This follows from Theorem 5.1 using the Stein factorization (see e.g. [Ha1]). Indeed, for a non-finite f, let f = g ∘ h be its Stein factorization, i.e. h : X → X′ has connected fibers, and g : X′ → ℙ^{N} is finite. Then dim X′ = dim f(X), therefore g^{−1}(L) is connected by Theorem 5.1, therefore f^{−1}(L) is also connected. ⊓ ⊔

Theorem 5.3 ([FH]) Let ℙ = ℙ^{N} × … × ℙ^{N} (r copies), and let Δ be the diagonal. If X is an irreducible variety, f : X → ℙ a morphism, with

dim f(X) > codim_{ℙ} Δ = (r − 1)N,

then f^{−1}(Δ) is connected.

Proof. Identify ℙ as the set of ordered r-tuples of points of ℙ^{N}, so that a point in ℙ has r factors in ℙ^{N}; the diagonal Δ consists of those points whose factors are all equal.

For simplicity suppose that f is a finite morphism, a general case is handled with a help of the Stein factorization as in the proof of the previous Corollary. Suppose that f^{−1}(Δ) is not connected, choose points x and x′ in different connected components of f^{−1}(Δ). Choose a hyperplane H in ℙ^{N} not containing any of the r factors of f(x) or f(x′). Let ℙ_{0} be the open set of ℙ consisting of those points none of whose factors lie in H. Then ℙ_{0} is a product of r copies of ℂ^{N}, which is identified with ℂ^{rN} and therefore with ℙ^{rN} \ {z_{0} = 0}, where z_{0}, …, z_{rN} are the homogeneous coordinates on ℙ^{rN}. Let W ⊂ ℙ × ℙ^{rN} be the closure of the graph of this birational correspondence. If coordinates are chosen on ℙ^{N} such that H is the hyperplane x_{0} = 0, and the homogeneous coordinates on the k-th copy of ℙ^{N} are x_{0}^{k}, …, x_{N}^{k}, then a point (x^{1}) × … × (x^{r}) × (z) in ℙ × ℙ^{rN} belongs to W if and only if there are constants λ_{1}, …, λ_{r} such that

(z_{0}, z_{(k−1)N+1}, …, z_{kN}) = λ_{k}(x_{0}^{k}, …, x_{N}^{k})

for all k = 1, …, r. So W is defined by the equations

\{ \begin{matrix}x_{j}^{k}z_{0} = x_{0}^{k}z_{(k−1)N+j} & 1 ≤ k ≤ r, 1 ≤ j ≤ N \\ x_{i}^{k}z_{(k−1)N+j} = x_{j}^{k}z_{(k−1)N+i} & 1 ≤ k ≤ r, 1 ≤ i < j ≤ N. \end{matrix}

Let α : W → ℙ, β : W → ℙ^{rN} be the two birational projections. Let L be the N-dimensional linear subspace of ℙ^{rN} defined by the equations

z_{(k−1)N+j} = z_{kN+j}, 1 ≤ k ≤ r − 1, 1 ≤ j ≤ N.

Then it is easy to check that

α^{−1}(Δ) ⊃ β^{−1}(L) and α^{−1}(Δ ∩ ℙ_{0}) ⊂ β^{−1}(L).  (5.1)

Let X̃ be the irreducible component of the fiber product X ×_{ℙ} W that maps onto X. Then there is a diagram

X̃ ⟶^{g} W ⟶^{β} ℙ^{rN}
π↓  ↓^{α}
X ⟶_{f} ℙ

where the square commutes, and π is surjective. Then equations (5.1) imply that

f^{−1}(Δ) ⊃ π(g^{−1}β^{−1}(L)), and f^{−1}(Δ ∩ ℙ_{0}) ⊂ π(g^{−1}β^{−1}(L)).  (5.2)

Now dim βg(X̃) = dim f(X) since f(X) meets the open set ℙ_{0} which corresponds isomorphically to an open set in ℙ^{rN}. Then by Theorem 5.1 we know that (βg)^{−1}(L) is connected. Therefore π((βg)^{−1}(L)) is connected. But by (5.2) π((βg)^{−1}(L)) is then a connected subset of f^{−1}(Δ) which contains both x and x′. Contradiction. ⊓ ⊔

Corollary 5.4 ([FH])

(a) Let Y be an irreducible subvariety of the projective space ℙ^{N} and let f : X → ℙ^{N} be a morphism from an irreducible variety X to ℙ^{N} with dim f(X) > codim_{ℙ^{N}} Y. Then f^{−1}(Y) is connected.

(b) If X and Y are irreducible subvarieties of ℙ^{N} with dim X + dim Y > N then X ∩ Y is connected.

Proof. (a) We apply Theorem 5.3 for the map F : X × Y → ℙ^{N} × ℙ^{N} given by F = (f, Id).

(b) Follows from (a). ⊓ ⊔

Corollary 5.5 ([FH])

(a) Let X be an irreducible variety of dimension n, f : X → ℙ^{N} an unramified morphism, with N < 2n. Then f is a closed embedding.

(b) Let X be an irreducible n-dimensional subvariety of ℙ^{N} with N < 2n. Then the algebraic fundamental group of X is trivial (i.e. X has no nontrivial étale coverings).

Proof. (a) If f : X → ℙ^{N} is unramified, consider the product mapping f × f : X × X → ℙ^{N} × ℙ^{N}. The unramified assumption is equivalent to the assertion that the diagonal Δ_{X} in X × X is open as well as closed in (f × f)^{−1}(Δ_{ℙ^{N}}). Since f is automatically finite, dim(f × f)(X × X) = 2n, so (f × f)^{−1}(Δ_{ℙ^{N}}) is connected by the Theorem 5.3. Hence

Δ_{X} = (f × f)^{−1}(Δ_{ℙ^{N}}),

so f is one-to-one, and therefore a closed embedding.

(b) Follows from (a). ⊓ ⊔

5.2 Secant and Tangential Varieties

Definition 5.6 Suppose that X ⊂ ℙ^{N} is a smooth n-dimensional projective variety. The tangential variety Tan(X) is the union of all (embedded) tangent spaces to X

Tan(X) = ∪_{x∈X} T̂_{x}X.

The secant variety Sec(X) is the closure of the union of all secant lines to X

Sec(X) = \overline{∪_{x,y∈X} ℙ_{xy}^{1}},

where ℙ_{xy}^{1} is a line connecting x and y.

Clearly, Tan(X) and Sec(X) are closed irreducible subvarieties of ℙ^{N}, with Tan(X) ⊂ Sec(X), and

dim Tan(X) ≤ 2n, dim Sec(X) ≤ 2n + 1.

Example 5.7 Let V = ℂ^{k+1} ⊗ ℂ^{l+1} denote the space of (k + 1) × (l + 1) matrices. Let X ⊂ ℙ(V) be the projectivization of the variety of rank 1 matrices. Then X is isomorphic to ℙ^{k} × ℙ^{l} embedded into ℙ(V) via the Segre embedding. It is easy to see that Sec(X) is equal to the projectivization of the variety of matrices of rank at most 2.

Theorem 5.8 ([FH], [Z2]) Suppose that X ⊂ ℙ^{N} is a smooth projective variety. Then either

dim Tan(X) = 2n, dim Sec(X) = 2n + 1

or

Tan(X) = Sec(X).

Proof. Indeed, suppose that Tan(X) ≠ Sec(X). Since Sec(X) is irreducible, it is sufficient to prove that in this case dim Tan(X) = 2n. Suppose, on the contrary, that dim Tan(X) < 2n. Let L ⊂ ℙ^{N} be a linear subspace such that L ∩ Tan(X) = ∅ and dim L = N − 1 − dim Tan(X) (since Tan(X) ≠ Sec(X) we obviously have dim Tan(X) ≤ N − 1). Let π be the linear projection ℙ^{N} → ℙ^{dim Tan(X)} with center L. Since L ∩ Tan(X) = ∅, the restriction of π on X is an unramified morphism. Since dim Tan(X) < 2n, it follows by Theorem 5.5 that the restriction of π on X is a closed embedding. Therefore, L ∩ Sec(X) = ∅. Since dim L + dim Sec(X) ≥ N, this is a contradiction. ⊓ ⊔

It is possible to generalize this theorem in the following direction. Suppose that Y ⊂ X ⊂ ℙ^{N} are arbitrary irreducible projective varieties. Then we can define relative secant and tangential varieties as follows. The relative secant variety Sec(Y, X) is the closure of the union of all secants ℙ_{x,y}^{1}, where x ∈ X, y ∈ Y. The relative tangential variety Tan(Y, X) is the union of tangent stars T_{y}^{⋆}X for y ∈ Y, where the tangent star T_{x}^{⋆}X for x ∈ X is the union of limit positions of secants ℙ_{x′,x′′}^{1}, where x′, x′′ ∈ X and x′, x′′ → x.

Example 5.9 Let V = ℂ^{k+1} ⊗ ℂ^{l+1} denote the space of (k + 1) × (l + 1) matrices. Let X^{r} ⊂ ℙ(V) be the projectivization of the variety of matrices of rank at most r. Then X^{r} ⊂ X^{r+1} and for r < s we have

Sec(X^{r}, X^{s}) = \{ \begin{matrix}X^{r+s} & if r + s ≤ min(k + 1, l + 1) \\ ℙ(V) & if r + s > min(k + 1, l + 1). \end{matrix}

The proof of the following theorem is analogous to the proof of Theorem 5.8.

Theorem 5.10 ([Z2]) Suppose that Y ⊂ X ⊂ ℙ^{N} are irreducible projective varieties. Then either

dim Tan(Y, X) = dim X + dim Y, dim Sec(Y, X) = dim X + dim Y + 1

or

Tan(Y, X) = Sec(Y, X).

One can go even further and define the join S(X, Y) of any two subvarieties X, Y ⊂ ℙ(V),

S(X, Y) = \overline{∪_{x∈X, y∈Y} ℙ_{xy}^{1}},

where the closure is not necessary if the two varieties do not intersect. For example, if X and Y are projective subspaces then

S(X, Y) = ℙ(Cone(X) + Cone(Y)).

An important result about joins is the following Terracini lemma.

Lemma 5.11 ([Te]) Let X, Y ⊂ ℙ(V) be irreducible varieties and let

x ∈ X, y ∈ Y, z ∈ ℙ_{xy}^{1}.

Then

T̂_{z}S(X, Y) ⊃ S(T̂_{x}X, T̂_{y}Y).

Moreover, if z is a generic point of S(X, Y) then equality holds.

If an algebraic group G acts on ℙ^{N} with finitely many orbits, then for any two orbits O_{1} and O_{2} the join of their closures S(\overline{O_{1}}, \overline{O_{2}}) is G-invariant therefore coincides with some orbit closure \overline{O}. This gives an interesting associative product on the set of G-orbits.

Example 5.12 Let L be a simple algebraic group and P a parabolic subgroup with abelian unipotent radical. In this case 𝔩 = Lie L admits a short ℤ-grading with only three non-zero parts:

𝔩 = 𝔩_{−1} ⊕ 𝔩_{0} ⊕ 𝔩_{1}.

Here 𝔩_{0}⊕𝔩_{1} = Lie P and exp(𝔩_{1}) is the abelian unipotent radical of P. Let G ⊂ L be a reductive subgroup with Lie algebra 𝔩_{0}. Recall that by Theorem 2.17 G has finitely many orbits in 𝔩_{1} naturally labeled by integers from the segment [0, r]. Let X_{i} = \overline{ℙ(O_{i})}, i = 0, …, r. Then it is easy to see that

S(X_{i}, X_{j}) = X_{min(i+j,r)}.

Indeed, by associativity and induction it suffices to prove that S(X_{i}, X_{1}) = X_{min(i+1,r)} and this easily follows from the results of Section 2.3.

5.3 Zak Theorems

The following theorem is called Zak Theorem on tangencies.

Theorem 5.13 ([Z2], [FL])

(a) Suppose that X ⊂ ℙ^{N} is a smooth nondegenerate projective variety, dim X = n. If L is a k-plane in ℙ^{N} and k ≥ n then

dim Sing(L ∩ X) ≤ k − n.

(b) If X ⊂ ℙ^{N} is a non-linear smooth projective variety, then dim X^{∗} ≥ X^{∗}. Moreover, if X^{∗} is smooth, then dim X = dim X^{∗}.

Proof. (a) Indeed, let Y = Sing L ∩ X. Then x ∈ Y if and only if T̂_{x}X ⊂ L. Therefore, Tan(Y, X) ⊂ L. In particular, dim Tan(Y, X) ≤ k. On the over hand, since X is non-degenerate, X ⊄ L. Therefore Sec(Y, X) ⊄ L.

In particular, Tan(Y, X) ≠ Sec(Y, X). By Theorem 5.10 it follows that dim Tan(Y, X) = dim Y + n. Finally, we have dim Y ≤ k − n.

(b) Let H ⊂ X^{∗}_{sm}. Then the contact locus Sing X ∩ H is the projective subspace of dimension def X. Therefore, by (a) we have

def X ≤ dim H − dim X = codim X − 1.

It follows that dim X ≤ dim X^{∗}. If X^{∗} is also smooth then using the reflexivity theorem and the same argument as above we get dim X^{∗} ≤ dim X, therefore dim X = dim X^{∗}. ⊓ ⊔

To remove the condition of smoothness in Theorem 5.13 (a) we need the notion of Gauss images. Let X ⊂ ℙ^{N} be an irreducible n-dimensional projective variety.

Definition 5.14 For any k ≥ n the variety

γ_{k}(X) = \overline{∪_{x∈X_{sm}}\{L ⊂ Gr(k, ℙ^{N}) | L ⊃ T̂_{x}X\}} ⊂ Gr(k, ℙ^{N})

is called the k-th Gauss image of X.

It is easy to see that Gauss images are irreducible varieties. For example, γ_{N−1}(X) ⊂ Gr(N − 1, ℙ^{N}) = ℙ^{N}^{∗} is equal to the dual variety X^{∗}.

Example 5.15 One might expect that usually dim X = dim γ_{n}(X). Indeed, suppose that dim X > dim γ_{n}(X). Then a generic fiber F of the map X_{sm} → γ_{n}(X), x ↦ T̂_{x}X has positive dimension dim X −dim γ_{n}(X). Therefore, if the hyperplane H is tangent to X_{sm} at some point x ∈ F then H is tangent to X at all points of F. It easily follows that def X > dim F = dim X −dim γ_{n}(X). So X has a positive defect in this case.

The proof of the following theorem is analogous to the proof of Theorem 5.13 (a).

Theorem 5.16 ([Z2]) Suppose that X ⊂ ℙ^{N} is a nondegenerate projective variety. If L ⊂ γ_{k}(X) then

dim{x ∈ X T̂_{x}X ⊂ L} ≤ k − n + (dim Sing X + 1),

where we set dim Sing X = −1 if X is smooth.

The following theorem is called Zak’s Theorem on linear normality.

Theorem 5.17 ([Z4]) Suppose that X ⊂ ℙ^{N} is a smooth non-degenerate projective variety, dim X = n. If

codim X < \frac{N + 4}{3}

then Sec(X) = ℙ^{N}.

Proof. By Theorem 5.8 either Tan(X) = Sec(X) or dim Sec(X) = 2n + 1. Suppose that Sec(X) ≠ ℙ^{N} and dim Sec(X) = 2n + 1. Then 2n + 1 < N, which contradicts codim X < \frac{N+4}{3}.

Suppose now that Tan(X) = Sec(X) and Sec(X) ≠ ℙ^{N}. Let z ∈ Sec(X) be a generic point, L = T̂_{z} Sec(X), and Q = {x ∈ X | z ∈ T̂_{x}X}. Then Q is closed and an easy dimension count shows that dim Q = 2n−dim Sec(X). For any point x ∈ Q we have z ∈ T̂_{x}X ⊂ Sec(X) therefore T̂_{x}X ⊂ T̂_{z} Sec(X) = L. It follows that Tan(Q, X) ⊂ L. However, Sec(Q, X) ⊄ L because L is a proper subspace and X is not degenerate. By Theorem 5.10 we have

dim Sec(Q, X) = dim Q + dim X + 1 = 3n − dim Sec(X) + 1.

Since Sec(Q, X) ⊂ Sec(X), we have dim Sec(Q, X) ≤ dim Sec(X). Then

2 dim Sec X ≥ 3n + 1

Hence 2N > 3n + 1. But this contradicts codim X < \frac{N+4}{3}. ⊓ ⊔

Theorem 5.17 is called the theorem on linear normality due to the following corollary first conjectured in [Ha2].

Corollary 5.18 Suppose that X ⊂ ℙ^{N} is a smooth non-degenerate projective variety. If

codim X < \frac{N + 2}{3}

then X is linearly normal.

Proof. Recall from Section 1.4 that if X is not linearly normal then there exists a subvariety X′ ⊂ ℙ^{N+1} and a point p ∈ ℙ^{N+1} such that X is isomorphic to X′ via the linear projection with center p. In particular, Sec(X′) ≠ ℙ^{N+1}. But this contradicts Theorem 5.17. ⊓ ⊔

Zak [Z2] has also classified the varieties in the borderline case

codim X = \frac{N + 4}{3}.

Theorem 5.19 ([Z4]) Suppose that X ⊂ ℙ^{N} is a smooth non-degenerate ndimensional projective variety such that codim X = \frac{N+4}{3}. Then Sec(X) = ℙ^{N} except the 4 following cases:

– n = 2, X = ℙ^{2}, X ⊂ ℙ^{5} is the Veronese embedding.

– n = 4, X = ℙ^{2} × ℙ^{2}, X ⊂ ℙ^{8} is the Segre embedding.

– n = 8, X = Gr(2, ℂ^{6}), X ⊂ ℙ^{14} is the Plücker embedding.

– n = 16, X ⊂ ℙ^{26} is the projectivization of the highest weight vector orbit in the 27-dimensional irreducible representation of E_{6}.

The varieties listed in Theorem 5.19 are called Severi varieties after Severi who proved that the unique 2-dimensional Severi variety is the Veronese surface. Scorza and Fujita–Roberts [FR] have shown that the unique 4dimensional Severi variety is the Segre embedding of ℙ^{2} × ℙ^{2}. It is also shown in [FR] that n ≡ 0 mod 16 as n > 8. Finally, Tango [Tan] proved that if n > 16 then either n = 2^{a} or n = 3 · 2^{b}, where a ≥ 7 and b ≥ 5. The 16-dimensional Severi variety was discovered by Lazarsfeld [La]. In fact, all Severi varieties (1)–(4) arise from Example 5.12. Another description of Severi varieties can be found in Example 2.3.

80 5. Fulton–Hansen Theorem and Applications

6. Dual Varieties and Projective Differential Geometry

Preliminaries

In sections 6.1 and 6.2 we review applications of the Katz dimension formula expressing the codimension of the dual variety in terms of the rank of a certain Hessian matrix. The section 6.3 contains results of Ein about the contact locus. In the last section 6.4 we give a useful formalism for dealing with dual varieties “under the microscope”.

6.1 The Katz Dimension Formula

Let X ⊂ ℙ^{n} = ℙ(V) be an irreducible k-dimensional projective variety and X^{∗} ⊂ ℙ^{n}^{∗} = ℙ(V^{∗}) be the projectively dual variety. The Katz dimension formula expresses the defect def(X) = codim X^{∗} − 1 in terms of the rank of a certain Hessian matrix.

For x ∈ ℙ(V) let x^{⊥} ⊂ V^{∗} denote the annihilator of the line in V corresponding to x. Let x_{0} be a smooth point of X. Then one can choose linear functionals

T_{0} ∈ V^{∗} \ x_{0}^{⊥}, T_{1}, …, T_{k} ∈ x_{0}^{⊥}

so that the functions

t_{1} = T_{1}/T_{0}, t_{2} = T_{2}/T_{0}, …, t_{k} = T_{k}/T_{0}

are local coordinates on X in the neighborhood of x_{0}. For every U ∈ x_{0}^{⊥} the function u = U/T^{0} on X near x_{0} is an analytic function of t_{1}, …, t_{k} such that u(0, …, 0) = 0. Consider the Hessian matrix

Hes(u) = Hes(U; T_{0}, T_{1}, …, T_{k}; x_{0}) = (\frac{∂^{2}u}{∂t_{i}∂t_{j}} (0, …, 0))_{i,j=1,...,k}.

Theorem 6.1 ([Ka]) Let def(X) = codim X^{∗} − 1 be the defect of X. Then

(a) We have def(X) = min corank Hes(u),

the minimum over all possible choices of x_{0} and U.

(b) X^{∗} is a hypersurface if and only if for some choice of x_{0} and u the Hessian matrix Hes(u) is invertible.

Proof. Let I_{X}^{0} ⊂ ℙ(V) × ℙ(V^{∗}) be a “smooth part” of the conormal variety consisting of all pairs (x, H) such that x ∈ X is smooth and H is tangent to X at x. Let π : I_{X}^{0} → ℙ(V^{∗}) be the second projection. Then X^{∗} is the closure of π(I_{X}^{0}). We shall compute the Jacobian matrix of π in appropriate local coordinates and relate it to the Hessian matrices appearing in the Theorem. Then we shall use an obvious fact that for any regular map π : Z → S of algebraic varieties the dimension of the image is equal to the maximal rank of the Jacobian matrix of π at smooth points of Z.

Let (x_{0}, H_{0}) ∈ I_{X}^{0}. Then one can choose

T_{0} ∈ V^{∗} \ x_{0}^{⊥}, T_{1}, …, T_{k} ∈ x_{0}^{⊥}

so that the functions

t_{1} = T_{1}/T_{0}, t_{2} = T_{2}/T_{0}, …, t_{k} = T_{k}/T_{0}

are local coordinates on X in the neighborhood of x_{0}. Extend T_{1}, …, T_{k} to a basis

{T_{1}, …, T_{k}, U_{1}, …, U_{n−k}}

of x_{0}^{⊥}. Then each of functions u_{i} = U_{i}/T_{0} on X is an analytic function of t_{1}, …, t_{k} near x_{0} = (0, …, 0), and we have

u_{i}(0, …, 0) = 0 for i = 1, …, n − k.

Let x ∈ X be a point close to x_{0} with local coordinates (t_{1}, …, t_{k}), so in homogeneous coordinates

x = (1 : t_{1} : …: t_{k} : u_{1} : …: u_{n−k}).

Let H ⊂ ℙ(V^{∗}) be the hyperplane defined by an equation

∑_{j=0}^{k} τ_{j}T_{j} + ∑_{i=1}^{n−k} η_{i}U_{i} = 0.

Then H is tangent to X at x if and only if the function

τ_{0} + ∑_{j=1}^{k} τ_{j}t_{j} + ∑_{i=1}^{n−k} η_{i}u_{i}

vanishes at x together with all its first derivatives. Therefore, for a given x, the hyperplanes tangent to X at x form a projective space of dimension n − k − 1 with homogeneous coordinates (η_{1} : …: η_{n−k}), and the remaining coordinates are given by

τ_{j} = − ∑_{i=1}^{n−k} η_{i} \frac{∂u_{i}}{∂t_{j}} , j = 1, …, k,

τ_{0} = − (∑_{j=1}^{k} τ_{j}t_{j} + ∑_{i=1}^{n−k} η_{i}u_{i}) = − ∑_{i=1}^{n−k} η_{i}u_{i} + ∑_{j=1}^{k} ∑_{i=1}^{n−k} η_{i}t_{j} \frac{∂u_{i}}{∂t_{j}} .

Without loss of generality we may assume that the hyperplane H_{0} has coordinates

(η_{1} : …: η_{n−k}) = (0 : …: 0 : 1).

It follows that we can set η_{n−k} = 1 in above formulas, and use

(t_{1}, …, t_{k}, η_{1}, …, η_{n−k−1})

as local coordinates on I_{X}^{0} near (x_{0}, H_{0}). In these coordinates the projection π : I_{X}^{0} → ℙ(V^{∗}) has a form

(t_{1}, …, t_{k}, η_{1}, …, η_{n−k−1}) ↦ (τ_{0}, τ_{1}, …, τ_{k}, η_{1}, …, η_{n−k−1}),

where τ_{i} are given by formulas above with η_{n−k} = 1. Therefore, the Jacobian matrix of π at the origin is the following matrix:

\frac{∂(τ_{0}, τ_{1}, …, τ_{k}, η_{1}, …, η_{n−k−1})}{∂(t_{1}, …, t_{k}, η_{1}, …, η_{n−k−1})}(0, …, 0) =

= ( \begin{matrix}0 & 0 \\ − Hes(u_{n−k}) & − (\frac{∂(u_{1},...,u_{n−k})}{∂(t_{1},...,t_{k})}(0, …, 0))^{t} \\ 0 & Id_{n−k−1} \end{matrix} ) .

Here vertical sizes of the blocks are 1, k, n − k − 1, and horizontal sizes are k, n − k − 1. Clearly, this matrix has rank

n − k − 1 + rk Hes(u_{n−k}).

It follows that

dim X^{∗} = n − k − 1 + max rk Hes(u),

and the Theorem is proved. ⊓ ⊔

If X ⊂ ℙ^{n} is a hypersurface (or a complete intersection) then it is possible to rewrite Hessian matrices in homogeneous coordinates. In case of hypersurfaces the corresponding result was first formulated by B. Segre [Se]:

Theorem 6.2 Let f(x_{0}, …, x_{n}) be an irreducible homogeneous polynomial and let X ⊂ ℙ^{n} be the hypersurface with the equation f = 0. Let m be the largest number with the following property: there exists (m × m)-minor of the Hessian matrix (∂^{2}f/∂x_{i}∂x_{j}) that is not divisible by f. Then dim X^{∗} = m − 2.

Example 6.3 An irreducible surface in ℙ^{3} with equation f(x_{1}, …, x_{3}) = 0 is projectively dual to a space curve if and only if the Hessian det |∂^{2}f/∂x_{i}∂x_{j}| is divisible by f.

The Segre Theorem is applicable in the most common situation: the projectively dual variety is usually a hypersurface, therefore by taking duals of hypersurfaces we can get a considerable amount of varieties that are not hypersurfaces.

Example 6.4 The following remark belongs to Zak (unpublished). In notations of Theorem 6.2, suppose that the Hessian H(f) = |∂^{2}f/∂x_{i}∂x_{j}| is not trivial.. Then H(f) is divisible by f^{n+1−m} (this result can be easily verified by induction using the well-known fact that if any (p × p)-minor of a (p+1)×(p+1)-matrix A over the UFD D is divisible by f^{l}, where f is prime, then det A is divisible by f^{l+1}). Let deg f = d. We can compare degrees of H(f) and f^{n+1−m} and obtain the formula (n + 1)(d − 2) ≥ d(n + 1 − m),

i.e. d ≥ \frac{2(n + 1)}{m}. In other words, if Y is a projective variety in ℙ^{n} such that

Y^{∗} is a hypersurface with the non-trivial Hessian, then

deg Y^{∗} ≥ \frac{2(n + 1)}{dim Y + 2}.

Zak conjectures that the non-triviality of the Hessian can be substituted for the non-degeneracy of Y and, moreover, that this inequality turns to the equality if and only if Y is the projectivization of the variety of rank 1 elements in a simple Jordan algebra.

6.2 Product Theorem and Applications

6.2.A Product Theorem

Let X_{1} ⊂ ℙ^{n_{1}} and X_{2} ⊂ ℙ^{n_{2}} be two irreducible projective varieties. The product X_{1} × X_{2} is then naturally embedded in ℙ^{n_{1}} × ℙ^{n_{2}} and the latter is embedded in ℙ^{n} via the Segre embedding, where

n + 1 = (n_{1} + 1)(n_{2} + 1).

Therefore we have an embedding X_{1} × X_{2} ⊂ ℙ^{n}. Let (X_{1} × X_{2})^{∗} ⊂ ℙ^{n}^{∗} be the dual variety. It turns out that it is quite easy to calculate def X_{1} × X_{2}.

Theorem 6.5 ([WZ])

def X_{1} × X_{2} = max(0, def X_{1} − dim X_{2}, def X_{2} − dim X_{1}).

Proof. We shall use Theorem 6.1. Let k_{1} = dim X_{1}, k_{2} = dim X_{2}. Let x_{0} = (x_{01}, x_{02}) be a smooth point of X_{1} × X_{2}. We choose local coordinates

t_{11}, t_{21}, …, t_{k_{1}1}

of X_{1} near x_{01} and

t_{12}, t_{22}, …, t_{k_{2}2}

of X_{2} near x_{02}. So for ν = 1, 2, a point x_{ν} close to x_{0ν} has homogeneous coordinates

x_{ν} = (1 : t_{1ν} : …: t_{k_{ν}ν} : u_{1ν} : …: u_{n_{ν}−k_{ν},ν}),

where each u_{1ν} is an analytic function of

t_{1ν}, …, t_{k_{ν}ν}

vanishing at the origin.

By definition of the Segre embedding, a point (x_{1}, x_{2}) ∈ X_{1} × X_{2} has as homogeneous coordinates all pairwise products of homogeneous coordinates of x_{1} and x_{2}. Therefore, the set of homogeneous coordinates (near (x_{01}, x_{02})) is given by

(1 : t_{1} : t_{2} : u_{1} : u_{2} : tt : tu : ut : uu),

where symbols

t_{1}, t_{2}, u_{1}, u_{2}, tt, tu, ut, uu

stand for the following sets of variables:

t_{1} = {t_{j1}}, t_{2} = {t_{j2}}, u_{1} = {u_{i1}}, u_{2} = {u_{i2}},
tt = {t_{j_{1}1}t_{j_{2}2}}, tu = {t_{j1}u_{i2}}, ut = {u_{i1}t_{j2}}, uu = {u_{i_{1}1}u_{i_{2}2}}.

The coordinates in sets t_{1} and t_{2} can be chosen as local coordinates on X_{1}×X_{2} near (x_{01}, x_{02}). The coordinates from remaining sets are analytic functions of these local coordinates vanishing at the origin.

To apply Theorem 6.1, we have to consider the Hessian matrix Hes(u), where u is a linear combination of all the coordinates from the sets

t_{1}, t_{2}, u_{1}, u_{2}, tt, tu, ut, uu

regarded as a function of local coordinates from t_{1} and t_{2}. We can write Hes(u) as

Hes(u) = ( \begin{matrix}A_{11} & A_{12} \\ A_{21} & A_{22} \end{matrix} ) ,

where

A_{αβ} = (\frac{∂u}{∂t_{iα}∂t_{jβ}} (0, …, 0))_{1≤i≤k_{α}, 1≤j≤k_{β}} .

Then it is easy to see that A_{11} = Hes(u_{1}), A_{22} = Hes(u_{2}), and

Hes(u) = ( \begin{matrix}Hes(u_{1}) & A \\ A^{⊥} & Hes(u_{2}) \end{matrix} ) ,

where A can be any (k_{1}×k_{2})-matrix. Therefore, by Theorem 6.1 it is sufficient to use the following fact from linear algebra:

Lemma 6.6 For any integers k_{1} ≥ c_{1} ≥ 0, k_{2} ≥ c_{2} ≥ 0, let Bil(c_{1}, k_{1}; c_{2}, k_{2}) denote the set of all bilinear forms on ℂ^{k_{1}} ⊕ ℂ^{k_{2}} such that their restrictions on ℂ^{k_{1}} and ℂ^{k_{2}} have coranks c_{1} and c_{2}. Then a generic form φ ∈ Bil(c_{1}, k_{1}; c_{2}, k_{2}) has corank

corank(φ) = max(0, c_{1} − k_{2}, c_{2} − k_{1}).

Let us prove this lemma. The corank of a bilinear form is equal to the dimension of its kernel. Let φ ∈ Bil(c_{1}, k_{1}; c_{2}, k_{2}), U = Ker φ|_{ℂ^{k_{1}}} , dim U = c_{1}. The form φ induces a linear map ψ : U → (ℂ^{k_{2}})^{∗}. Let U_{0} = Ker ψ. Then it is clear that U_{0} ⊂ Ker φ and dim U_{0} ≥ dim U − dim(ℂ^{k_{2}})^{∗} = c_{1} − k_{2}. This argument shows that

corank(φ) ≥ max(0, c_{1} − k_{2}, c_{2} − k_{1}).

It remains to find a form such that this inequality turns to an equality. We shall proceed by induction. Suppose first that c_{1} and c_{2} are positive. Take a form φ′ ∈ Bil(c_{1} − 1, k_{1} − 1; c_{2} − 1, k_{2} − 1) such that

corank(φ′) = max(0, c_{1} − k_{2}, c_{2} − k_{1}).

We can define the form φ on

ℂ^{k_{1}} ⊕ ℂ^{k_{2}} = (ℂ^{k_{1}−1} ⊕ ⟨e⟩) ⊕ (ℂ^{k_{2}−1} ⊕ ⟨f⟩)

such that its restriction on ℂ^{k_{1}−1}⊕ℂ^{k_{2}−1} coincides with φ′, subspaces ℂ^{k_{1}−1}⊕ ℂ^{k_{2}−1} and ⟨e, f⟩ are orthogonal, and φ(e, e) = φ(f, f) = 0, φ(e, f) = 1. Then it is clear that φ ∈ Bil(c_{1}, k_{1}; c_{2}, k_{2}) and

corank(φ) = corank(φ′) = max(0, c_{1} − k_{2}, c_{2} − k_{1}).

Now we may assume that c_{1} = 0. Suppose that c_{2} and k_{1} are both positive. Take a form φ′ ∈ Bil(0, k_{1} − 1; c_{2} − 1, k_{2} − 1) such that

corank(φ′) = max(0, c_{2} − k_{1}).

We define the form φ on

ℂ^{k_{1}} ⊕ ℂ^{k_{2}} = (ℂ^{k_{1}−1} ⊕ ⟨e⟩) ⊕ (ℂ^{k_{2}−1} ⊕ ⟨f⟩)

such that the restriction of φ on ℂ^{k_{1}−1} ⊕ ℂ^{k_{2}−1} coincides with φ′, subspaces ℂ^{k_{1}−1} ⊕ℂ^{k_{2}−1} and ⟨e, f⟩ are orthogonal, and φ(e, e) = φ(e, f) = 1, φ(f, f) = 0. It is clear then that φ ∈ Bil(0, k_{1}; c_{2}, k_{2}) and

corank(φ) = corank(φ′) = max(0, c_{2} − k_{1}).

It remains to consider only cases when either k_{1} = 0 or c_{1} = c_{2} = 0. In both cases lemma is obvious. ⊓ ⊔

It follows immediately from Theorem 6.5 that the same result holds for the product of any number of factors. Let X_{k} ⊂ ℙ^{n_{k}}, k = 1 …, r be irreducible projective varieties. The product X = X_{1} ×…×X_{r} is then naturally embedded in ℙ^{n_{1}} × … × P^{n_{r}} and the latter is embedded in ℙ^{n} via the Segre embedding, where n + 1 = (n_{1} + 1) × … × (n_{r} + 1). Therefore we have an embedding X ⊂ ℙ^{n}. Let X^{∗} ⊂ ℙ^{n}^{∗} be the dual variety.

Theorem 6.7

def X + dim X = max(dim X, def X_{1} + dim X_{1}, …, def X_{r} + dim X_{r}).

In particular, X^{∗} is a hypersurface if and only if

def X_{k} + dim X_{k} ≤ dim X, k = 1 …, r.

6.2.B Hyperdeterminants

For i = 1, …, r consider a projective space ℙ^{l_{i}} = ℙ(ℂ^{l_{i}+1}). The Segre embedding ℙ^{l_{1}} × … × ℙ^{l_{r}} ⊂ ℙ^{l}, ℙ^{l} = ℙ(ℂ^{l_{i}+1} ⊗ … ⊗ ℂ^{l_{r}+1}), identifies the affine cone Cone(ℙ^{l_{1}} × … × ℙ^{l_{r}}) with a variety of decomposable tensors in ℂ^{l_{i}+1} ⊗ … ⊗ ℂ^{l_{r}+1}. The corresponding discriminant (if exists) is called a hyperdeterminant for the matrix format (l_{1} + 1) × … × (l_{r} + 1), cf. Example 2.18. Theorem 6.5 allows to get a simple criterion for the existence of hyperdeterminants:

Theorem 6.8 The dual variety (ℙ^{l_{1}} × … × ℙ^{l_{r}})^{∗} is a hypersurface (and, hence, defines a hyperdeterminant) if and only if

2l_{k} ≤ l_{1} + … + l_{r}

for k = 1, …, r.

Proof. Since the dual variety of ℙ^{n}^{∗} is an empty set, def ℙ^{n} = n. Now everything follows from Theorem 6.5. ⊓ ⊔ 6.2.C Associated Hypersurfaces

Let X ⊂ ℙ(V) be an irreducible subvariety, ℙ^{l} = ℙ(U). By Theorem 6.5, the dual variety (X × ℙ^{l})^{∗} is a hypersurface in ℙ(V ⊗ U)^{∗} for def X ≤ l ≤ dim X. Since we always have def X ≤ dim X, we obtain a system of hypersurfaces naturally associated to X. Their interpretation was found in [WZ].

We can identify ℙ(V ⊗ U)^{∗} with the projectivization of the space of linear maps Hom(U, V^{∗}). By some abuse of notation we shall denote a non-zero linear map and its projectivization by the same letter. Let ℙ(V ⊗ U)_{0}^{∗} be an open subset consisting of operators having maximal possible rank l + 1. For any f ∈ ℙ(V ⊗ U)_{0}^{∗} we can associate two projective subspaces

U(f) = ℙ(Im(f)) ⊂ ℙ(V)^{∗}

and the orthogonal subspace U(f)^{⊥} ⊂ ℙ(V). Recall that I_{X}^{0} ⊂ ℙ(V) × ℙ(V)^{∗} is an open part of the conormal variety consisting of pairs (x, H) such that x ∈ X_{sm} and H is the hyperplane tangent to X at x.

Theorem 6.9 ([WZ]) The dual variety (X × ℙ^{l})^{∗} is the closure of the set of points f ∈ ℙ(V ⊗ U)_{0}^{∗} such that

(U(f)^{⊥} × U(f)) ∩ I_{X}^{0} ≠ ∅.

In two extreme cases l = dim X or l = def X this result can be made more precise. To go further, we need a notion of an associated hypersurface. Let X ⊂ ℙ^{n} be a k-dimensional irreducible subvariety. Consider the set Z(X) of all (n − k − 1)-dimensional projective subspaces in ℙ^{n} that intersect X. This is a subvariety in the Grassmanian Gr(n − k, n + 1) parametrizing all (n − k − 1)-dimensional projective subspaces in ℙ^{n}.

Theorem 6.10 The subvariety Z(X) is an irreducible hypersurface, called an associated hypersurface of X.

Proof. Let B(X) ⊂ X × Z(X) be an incidence variety consisting of pairs (x, L) such that x ∈ X, L ∈ Gr(n − k, n + 1), and x ∈ L. We have two projections p : B(X) → X and q : B(X) → Z(X). Then p is a Grassmannian fibration, any fiber of p is isomorphic to Gr(n − k − 1, n). In particular, B(X) (and hence Z(X)) is irreducible. On the other hand, q is birational, because a generic (n − k − 1)-dimensional projective subspace intersecting X meets X in exactly one point. Now an easy dimension count shows that Z(X) is a hypersurface. ⊓ ⊔

Now we can relate associated hypersurfaces and dual varieties. We continue to use the notation of Theorem 6.9. The following theorem is sometimes called a Cayley trick:

Theorem 6.11 ([WZ]) Let k = dim X, k^{∗} = dim X^{∗}. Then (a) Let l = dim X, and p_{1} : ℙ(V ⊗ U)_{0}^{∗} → Gr(n − k − 1, n + 1) be the map

f ↦ U(f)^{⊥}. Then

(X × ℙ^{l})^{∗} = \overline{p_{1}^{−1}(Z(X))}.

(b) Let l = def X, and p_{2} : ℙ(V ⊗ U)_{0}^{∗} → Gr(n − k^{∗} − 1, n + 1) be the map

f ↦ U(f). Then

(X × ℙ^{l})^{∗} = \overline{p_{2}^{−1}(Z(X^{∗}))}.

We refer the reader to [GKZ2] for further results on associated varieties and Chow forms.

6.3 Ein Theorems

Suppose that X ⊂ ℙ^{N} is a non-degenerate smooth projective variety, dim X = n. For any hyperplane H ⊂ ℙ^{N} the contact locus Sing X ∩ H is a subvariety of X consisting of all points x ∈ X such that the embedded tangent space T̂_{x}X is contained in H. One can use the Jacobian ideal of X ∩H to define a scheme structure on the contact locus, however this scheme could be not reduced. Clearly the contact locus is non-empty if and only if H belongs to the dual variety X^{∗}. By Theorem 1.12 if def X = d then for any H ∈ X^{∗}_{sm} the contact locus Sing(H ∩ X) is a projective subspace of dimension d and a union of this projective subspaces is dense in X. In this section we study further properties of the contact locus, most results here belong to L. Ein.

Theorem 6.12 ([E1]) Suppose that q is a generic point of X and H is generic tangent hyperplane of X at q, L = ℙ^{d} is the contact locus of H with X. Then

(a) if p is a point in ℙ^{d}, then the tangent cone of the hyperplane section H ∩ X at p is a quadric hypersurface of rank n − d in T̂_{p}(X).

(b) Let s_{h} be the section of O_{X} ⊗ O_{X}(1) defining H ∩ X. Then s_{h} factors through I_{L}^{2}, where I_{L} ⊂ O_{X} is the ideal sheaf of L in X.

(c) Let t_{h} be the section of I_{L}^{2}/I_{L}^{3}⊗O_{L}(1) ≅ S^{2}(N_{L}^{∗}X)⊗O_{L}(1) induced by s_{h}. Then t_{h} defines a nonsingular quadric hypersurface in ℙ(N_{L}^{∗}X|_{p}).

Proof. (a) This follows from the proof of Theorem 6.1 using Sard’s Lemma for algebraic varieties.

(b) We choose a local coordinate system {x_{1}, …, x_{n}} of X near p. We may assume that I_{L} is generated by x_{1}, …, x_{n−d}. Since L ⊂ H ∩ X, we can write the power series of s_{h} in the following form:

s_{h} = x_{1}f_{1} + … + x_{n−d}f_{n−d} + ∑_{i=1}^{n−d} ∑_{j=1}^{n−d} x_{i}x_{j}g_{ij},

where f_{1}, …, f_{n−d} are power series in variables x_{n−d+1}, …, x_{n} only. But

Sing H ∩ X = L, therefore \frac{∂s_{h}}{∂x_{i}} |_{L} = 0 for i = 1, …, n − d. Hence

f_{1} = …, f_{n−d} = 0

and

s_{h} = ∑_{i=1}^{n−d} ∑_{j=1}^{n−d} x_{i}x_{j}g_{ij}.

Thus s_{h} factors through I_{L}^{2}.

(c) We can write g_{ij} = a_{ij} + h_{ij}, where a_{ij}’s are constants and h_{ij}’s are power series without the constant term. Now

s_{h} = ∑_{i=1}^{n−d} ∑_{j=1}^{n−d} x_{i}x_{j}(a_{ij} + h_{ij}).

By (a), ∑_{i=1}^{n−d} ∑_{j=1}^{n−d} x_{i}x_{j}a_{ij} is a quadratic form of rank n − d. But this is also the

equation for the quadric hypersurface in ℙ(I_{L}/I_{L}^{2}) induced by s_{h}. ⊓ ⊔

Recall that any vector bundle on ℙ^{1} has a form ⊕_{i}O(a_{i}) for some integers a_{i} (see e.g. [Ha1]).

Definition 6.13 A vector bundle E on a projective space ℙ^{N} is called uniform if for any line T ⊂ ℙ^{N} the restriction E|_{T} is a fixed vector bundle ⊕_{i}O(a_{i}).

The following theorem is the main result of this section.

Theorem 6.14 ([E1]) Suppose that X is a nonlinear smooth projective variety in ℙ^{N}, def X = d > 0, q is a generic point of X and H is generic tangent hyperplane of X at q, L = ℙ^{d} is the contact locus of H with X. Then

(a) N_{L}X ≅ (N_{L}X)^{∗}(1).

(b) If T = ℙ^{1} is a line in L = ℙ^{d}, then

N_{L}X|_{T} ≅ O_{T}^{⊕(n−d)/2} ⊕ O_{T} (1)^{⊕(n−d)/2}

(in particular N_{L}X is a uniform vector bundle) and

N_{T} X ≅ O_{T}^{⊕(n−d)/2} ⊕ O_{T} (1)^{⊕(n+d−2)/2}.

(c) There is an irreducible \frac{3n + d − 4}{2}-dimensional family of lines in X. If

p is a generic point in X, then there is an \frac{n + d − 2}{2}-dimensional family

of lines in X through p.

Proof. (a) We continue to use the notation of Theorem 6.12. By this theorem, the section s_{h} gives a section of

I_{L}^{2}/I_{L}^{3}(1) = S^{2}(N_{L}^{∗}X)(1) ⊂ Hom(N_{L}X, N_{L}^{∗}X(1)).

Therefore s_{h} induces a map from N_{L}(X) to N_{L}X(1). This map is an isomorphism by Theorem 6.12 (c).

(b) Consider the exact sequence

0 → N_{L}X|_{T} → N_{L}ℙ^{N}|_{T} → N_{X}ℙ^{N}|_{T} → 0,

where N_{L}ℙ^{N}|_{T} = (N − d)O_{T} (1). Suppose that

N_{L}X|_{T} = ⊕_{i=1}^{n−d} O_{T} (a_{i}).

Then exactness implies that all a_{i} ≤ 1. Using the isomorphism between N_{L}X and N_{L}^{∗}X(1) we observe that each a_{i} ≥ 0. Therefore,

N_{L}X|_{T} ≅ O_{T}^{⊕(n−d)/2} ⊕ O_{T} (1)^{⊕(n−d)/2}

and N_{L}X is a uniform vector bundle. This implies that

N_{T} X ≅ O_{T}^{⊕(n−d)/2} ⊕ O_{T} (1)^{⊕(n+d−2)/2}.

The statement (c) is a consequence of the standard deformation theory. Indeed, let p ∈ L and let T_{0} be a line in L through p. Since

N_{T_{0}}X ≅ O_{T_{0}}^{⊕(n−d)/2} ⊕ O_{T_{0}}(1)^{⊕(n+d−2)/2},

we have

dim H^{0}(T_{0}, N_{T_{0}}X) = \frac{n − d}{2} + 2\frac{n + d − 2}{2} = \frac{3n + d − 4}{2}

and

H^{1}(T_{0}, N_{T_{0}}X) = 0.

Therefore, the Hilbert scheme of lines in X is smooth at the point T_{0} and there is a unique irreducible component of the Hilbert scheme containing the point T_{0}. This component has dimension (3n + d − 4)/2. The proof of the second statement of (c) is similar. ⊓ ⊔

As a quite formal consequence of Theorem 6.14 (b) we get the following parity theorem that was first proved by A. Landman, using the Picard– Lefschetz theory (unpublished):

Theorem 6.15 ([E1]) If def X > 0 then dim X ≡ def X mod 2.

Example 6.16 ([GH]) Suppose that X ⊂ ℙ^{n} is a smooth non-linear surface. Then X^{∗} is a hypersurface. Indeed, if def X > 0 then def X is even by Theorem 6.15. But X obviously can not contain an even-dimensional projective subspace ℙ^{def X}.

Example 6.17 Suppose that X ⊂ ℙ^{N} is a smooth projective nonlinear variety, dim X ≥ 2. Then def X ≤ dim X − 2. Indeed, it is clear that def X ≤ dim X − 1. Therefore, by Theorem 6.15 we have def X ≤ dim X − 2. Moreover, it can be shown [E1] that def X = dim X − 2 if and only if X is a projective bundle over a curve C, X = ℙ_{C}(F), where F is a rank n vector bundle on C and all fibers are embedded into ℙ^{N} linearly (c.f. Theorem 4.6 and remarks after it).

Example 6.18 Suppose that X ⊂ ℙ^{N} is a smooth projective nonlinear variety, dim X ≥ 3, N = 2n − 1, and dim X = dim X^{∗}. Since def X = n − 2 in this case, by Example 6.17 we have that X is a projective bundle over a curve C, X = ℙ_{C}(F), where F is a rank n vector bundle on C and all fibers are embedded into ℙ^{N} linearly. Moreover, by a theorem of S. Kleiman [Kl5] in this case X is the Segre embedding of ℙ^{1} × ℙ^{n−1}.

It was first observed by Griffiths and Harris [GH] that any smooth projective variety with positive defect has the negative Kodaira dimension, H^{0}(X, K_{X}^{⊗m}) = 0 for m > 0, where K_{X} is the canonical line bundle. This result will have important consequences in Chapter 9. Here is the detailed story:

Theorem 6.19 ([E1]) Suppose that X is a nonlinear smooth projective variety in ℙ^{N}, def X = d > 0, q is a generic point of X and H is a generic tangent hyperplane of X at q, L = ℙ^{d} is the contact locus of H with X. Then

(a) K_{X}|_{L} = O_{L} (\frac{−n − d − 2}{2}).

(b) The Kodaira dimension of X is negative.

(c) If K_{X} = O_{X}(a), then a = \frac{−n − d − 2}{2}.

(d) If n > \frac{N}{2} + 1, then K_{X} = O_{X} (\frac{−n − d − 2}{2}).

Proof. (a) By Theorem 6.14 (b), if T = ℙ^{1} is a line in L = ℙ^{d}, then

N_{L}X|_{T} ≅ O_{T}^{⊕(n−d)/2} ⊕ O_{T} (1)^{⊕(n−d)/2}.

It follows that Λ^{n−d}N_{L}X|_{T} = O_{T} (\frac{n − d}{2}) and, hence, Λ^{n−d}N_{L}X =

O_{L} (\frac{n − d}{2}). Therefore K_{X}|_{L} = O_{L} (\frac{−n − d − 2}{2}) by the adjunction for-

mula.

(b) Since there is such a plane L through a generic point p ∈ X, the linear system |K_{X}^{m}| is empty for m ≥ 0.

(c) follows from (a).

(d) follows from (c). Indeed, if dim X > \frac{N}{2} + 1, then Pic X = ℤ is gener-

ated by O_{X}(1) by the Barth Theorem [Ba]. ⊓ ⊔

Example 6.20 ([E1]) Suppose that X is a nonlinear smooth projective variety in ℙ^{N}, codim X = 2. Then X^{∗} is a hypersurface, unless X is the Segre embedding of ℙ^{1} ×ℙ^{2} in ℙ^{5}. Indeed, assume first that dim X ≥ 4 and def X =

d > 0. Then by Theorem 6.19 (d) the canonical class K_{X} = O_{X}(\frac{−n − d − 2}{2}).

However, it follows from results of [BC] that in this case X is automatically a complete intersection. This contradicts Theorem 4.2. Now, if dim X = 1 or dim X = 2 then X^{∗} is a hypersurface by Example 1.13 and Example 6.16. The case dim X = 3 is ruled out by Example 6.18.

In case of not generic contact loci the following version of Theorem 6.19 holds:

Theorem 6.21 ([BFS1]) Suppose that X is a smooth n-dimensional projective variety in ℙ^{N}, def X = d > 0. Let H ⊂ X^{∗}. Then Sing H ∩ X is the

union of projective subspaces ℙ^{d} with K_{X}|_{ℙ^{d}} = O_{ℙ^{d}} (\frac{−n − d − 2}{2}).

Proof. Let I_{X} ⊂ X × ℙ^{N}^{∗} be the conormal variety, π : I_{X} → ℙ^{N}^{∗} be the projection. Then π(I_{X}) = X^{∗}. Let p : I_{X} → X be the first projection, then p is a projective bundle. Clearly, I_{X} is smooth and irreducible. The general fibers F of π are isomorphic to the contact loci of X with generic tangent hyperplanes, therefore, they are isomorphic to the projective subspaces ℙ^{d} and by Theorem 6.19 we have p^{∗}(K_{X} + ((n + d)/2 + 1)L)_{F} ≃ O_{F}. It thus follows that every fiber of π is a union of projective subspaces ℙ^{d} with p^{∗}(K_{X} + ((n + d)/2 + 1)L)_{ℙ^{d}} ≃ O_{ℙ^{d}}. ⊓ ⊔

Let X be a non-linear n-dimensional smooth projective variety in ℙ^{N} such that def X = d > 0. Let H be a generic tangent hyperplane of X. Then the contact locus Sing H ∩X of X with H is a d-dimensional projective subspace L. Let X̃ be a blowing up of X along L. Let p : X̃ → X be the corresponding map. Denote by E the exceptional divisor and denote by F the proper transform of H ∩ X. Therefore, we have a following diagram

ℙ(N_{L}X) = E ⊂ X̃ ⊃  F
↓ ↓ ↓
L ⊂ X ⊃ H ∩ X.

We shall denote by OX̃(a, b) the line bundle p^{∗}O_{X}(a) ⊗ OX̃(−bE). For example, OX̃(F) = OX̃(1, 2).

The following vanishing theorem holds.

Theorem 6.22 ([E1]) Assume that K_{X} = O_{X}(b) for some b ∈ ℤ. Then

(a) K_{X} = O_{X} (\frac{−n − d − 2}{2}).

(b) KX̃ = OX̃ (\frac{−n − d − 2}{2}, −n + d + 1).

(c) H^{i}(OX̃(a, 1)) = 0, if i > 0 and a ≥ \frac{n − 3d − 2}{2}.

(d) H^{i}(OX̃(a, 2)) = 0, if i > 0 and a ≥ \frac{n − 3d}{2}.

Assume further that \frac{n − 3d − 2}{2} ≤ 0. Then

(e) H^{0}(N_{L}^{∗}X(a)) = 0 for a ≤ 0.

(f) H^{k}(N_{L}^{∗}X(a)) = 0 for a ≥ −d.

(g) H^{i}(N_{L}^{∗}X(a)) = 0 if 0 < i < d and a ≥ \frac{n − 3d}{2}.

(h) H^{i}(N_{L}^{∗}X(a)) = 0 if 0 < i < d and a ≤ \frac{d − n}{2}.

Proof. (a) follows from Theorem 6.19 (a).

(b) follows from (a) and the fact that X̃ is the blowing up of X along L.

(c) Let f : X̃ → ℙ^{N−1−d} be the projection with center L. Let Y = f(X̃). The hyperplane section H∩X will correspond to a hyperplane section D of Y. Then f^{−1}(D) = E +F and f^{∗}O_{Y} (1) = OX̃(1, 1). Let y ∈ Y \D, Z = f^{−1}(y). Suppose that dim Z ≥ 1. Since Z ∩ (E ∪ F) = ∅, p maps Z isomorphically to a variety in X. Therefore, OX̃(0, 1)|_{Z} is non-trivial. But OX̃(1, 1)|_{Z} = f^{∗}O_{Y} (1)|_{Z} is trivial. So OX̃(0, 1) is non-trivial. Hence Z ∩ E ≠ ∅, which is a contradiction. Therefore, all positive-dimensional fibers of f belong to E ∩F, in particular dim Y = dim X. Now,

OX̃(a, 1) = KX̃ ⊗ f^{∗}O_{Y} (n − d) ⊗ OX̃ (a − \frac{n − 3d − 2}{2}, 0) .

It follows from the Grauert–Rimenschneider vanishing theorem [GR] that

H^{i}(OX̃(a, 1)) = 0 if i > 0 and a ≥ \frac{n − 3d − 2}{2}.

(d) The proof is similar to (c).

(e) Consider the exact sequence

0 = H^{0}(OX̃(0, 1)) → H^{0}(O_{E}(0, 1)) → H^{1}(OX̃(0, 2)).

Now H^{1}(OX̃(0, 2)) = 0 by (d). So

H^{0}(O_{E}(0, 1)) ≃ H^{0}(N_{L}^{∗}X) = 0.

Hence H^{0}(N_{L}^{∗}X(a)) = 0 for a ≤ 0.

(f) Recall that N_{L}X = N_{L}^{∗}X(1). So (f) follows from (e) and Serre duality.

(g) Consider the exact sequence

H^{i}(OX̃(a, 1)) → H^{i}(O_{E}(a, 1)) → H^{i+1}(OX̃(a, 2)).

By (c) and (d), we conclude that

H^{i}(O_{E}(a, 1)) ≃ H^{i}(N_{L}^{∗}X(a)) = 0

for a ≥ \frac{n − 3d}{2}.

(h) follows from (g) and Serre duality. ⊓ ⊔

Theorem 6.23 ([E1]) Suppose that X is a nonlinear smooth projective variety in ℙ^{N}, dim X = n ≥ 3. Assume that K_{X} = O_{X}(b) for some b ∈ ℤ. Then

(a) def X ≤ \frac{n − 2}{2}.

(b) If dim X = 4m + 2 and def X = 2m > 0 then

N_{L}^{∗}X = H^{m} (N_{L}^{∗}X(−m)) ⊗ Ω_{L}^{m}(m),

and m ≤ 2.

Proof. Consider the Beilinson spectral sequence [OSS] with

E_{1}^{pq} = H^{q} (N_{L}^{∗}X(p)) ⊗ Ω_{L}^{−p}(−p),

which converges to

E^{i} = \{ \begin{matrix}N_{L}^{∗}X & if i = 0 \\ 0 & otherwise (i.e. E_{∞}^{pq} = 0 if p + q ≠ 0). \end{matrix}

(a) If k = def X ≥ \frac{n − 1}{2}, then \frac{n − 3k}{2} − 1 ≤ \frac{k − n}{2}. It follows from

Theorem 6.22 that H^{q}(N_{L}^{∗}X(p)) = 0 for −k ≤ p ≤ 0. It follows that N_{L}^{∗}X = 0. This is a contradiction.

(b) In this case H^{q}(N_{L}^{∗}X(p)) = 0 for −2m ≤ p ≤ 0 unless p = −m. This implies that N_{L}^{∗}X = H^{m}(N_{L}^{∗}X(−m)) ⊗ Ω_{L}^{m}(m). So

2m + 2 = rank N_{L}^{∗}X ≥ rank Ω_{L}^{m}(m) = \binom{2m}{m}.

It follows that m ≤ 2. ⊓ ⊔

We conclude this section with the following simple but very useful result first appeared in [LS] (where it was attributed to the referee). It could be called the monotonicity theorem.

Theorem 6.24 ([LS]) Let X ⊂ ℙ^{N} be a smooth projective n-dimensional variety. Suppose that through its generic point there passes a smooth subvariety Y of dimension h and defect θ. Then def X ≥ θ − k + h. In other words,

dim X + def X ≥ dim Y + def Y.

Proof. Suppose that x ∈ X is a generic point and H is a generic hyperplane tangent to X at x. Let Y be an h-dimensional submanifold passing through x such that def Y = θ. Since the defect is the dimension of a generic contact locus, which is a projective subspace, there exists a θ-dimensional projective subspace Z ⊂ Y containing x such that H is tangent to Y along Z (though H is not a generic tangent hyperplane to Y). Let f = 0 be a local equation for H ∩ X at x. In a neighborhood U of x, the differential df annihilates the tangent spaces T_{x}X and T_{z}Y for every z ∈ Z ∩ U. Hence df defines on Z ∩ U a section of the conormal bundle N_{Y}^{∗} X|_{Z} vanishing at x. Since N_{Y}^{∗} X is a (n − h)-dimensional vector bundle, it follows that this section in fact vanishes on a subvariety Z′ ⊂ Z ∩ U of codimension less then or equal to n − h. Therefore, H is tangent to X along Z′. But Sing X ∩ H is (def X)- dimensional, therefore def X ≥ θ − (n − h). ⊓ ⊔

6.4 The Projective Second Fundamental Form

6.4.A Gauss Map

Suppose that X ⊂ ℙ(V) is a smooth irreducible n-dimensional variety, not necessarily closed. Therefore, for any x ∈ X we have an embedded tangent space T̂_{x}X. A natural way to keep track of the motion of T̂_{x}X (while x moves in X) is the Gauss map.

Definition 6.25 The Gauss map γ is defined as follows.

γ : X → Gr(n, ℙ(V)), x ↦ T̂_{x}X,

where Gr(n, ℙ(V)) = Gr(n + 1, V) is the Grassmanian of n-dimensional projective subspaces in ℙ^{N}.

To measure how T̂_{x}X moves to the first order, one calculates the differential of γ

(dγ)_{x} : T_{x}X → T_{T̂_{x}X} Gr(n, ℙ(V)) = Hom(Cone(T_{x}X), V/ Cone(T_{x}X)).

It is easy to see that Cone(x) ⊂ Ker(dγ)_{x}(v) for any v ∈ T_{x}X, therefore d(γ)_{x} factors to a map

𝔽𝔽_{x}^{2} : T_{x}X → Hom(T_{x}X, N_{X}ℙ(V)|_{x}).

Moreover, 𝔽𝔽_{x}^{2} is symmetric, essentially because the Gauss map is already the derivative of a map and mixed partial derivatives commute. Finally, we get the following definition.

Definition 6.26 The section 𝔽𝔽^{2} of the vector bundle S^{2}T^{∗}X ⊗ N_{X}ℙ(V) constructed above is called the projective second fundamental form of X.

If X ⊂ ℙ(V) is a singular variety then 𝔽𝔽^{2} is still defined over the smooth locus X_{sm}.

It is convenient to consider 𝔽𝔽^{2} as a map 𝔽𝔽^{2} : N_{X}^{∗}ℙ(V) → S^{2}T^{∗}X and to set |𝔽𝔽^{2}| = ℙ(𝔽𝔽^{2}(N_{X}^{∗}ℙ(V))). One can think of |𝔽𝔽^{2}|_{x} as a linear family of quadric hypersurfaces in ℙ(T_{x}X). The space ℙ(N_{X}^{∗}ℙ(V)|_{x}) has the geometric interpretation as the space of hyperplanes tangent to X at x, i.e., the hyperplanes H such that X ∩H is singular at x. Then 𝔽𝔽_{x}^{2} can be viewed as the map that sends a hyperplane to the quadratic part of the singularity of X ∩ H at x.

Let Base |𝔽𝔽^{2}|_{x} ⊂ ℙ(T_{x}X) denote the variety of directions tangent to the lines that osculate to order 2 at x (that is, X appears to contain these lines to a second order). More precisely,

Base |𝔽𝔽^{2}|_{x} = ℙ{v ∈ T_{x}X | 𝔽𝔽_{x}^{2}(v, v) = 0}
= {p ∈ ℙ(T_{x}X) | p ∈ Q, ∀Q ∈ |𝔽𝔽^{2}|}.  (6.1)

Let Sing |𝔽𝔽^{2}|_{x} denote the set of tangent directions such that the embedded tangent space does not move to first order in these directions.

Sing |𝔽𝔽^{2}|_{x} = ℙ{v ∈ T_{x}X | 𝔽𝔽^{2}(v, w) = 0 ∀w ∈ T_{x}X}
= {p ∈ ℙ(T_{x}X) | p ∈ Q_{sing} ∀Q ∈ |𝔽𝔽^{2}|}.

The following theorem may be called the smoothness principle.

Theorem 6.27 ([GH]) Let X ⊂ ℙ^{N} be an irreducible projective variety. If X is smooth then Sing |𝔽𝔽^{2}|_{x} = ∅ for generic x ∈ X.

In other words, if X is smooth then the Gauss map γ is generically finite. In fact, much more is true.

Theorem 6.28 ([Z3, E4, Ra3]) If X is smooth then the Gauss map is finite and birational.

These theorem do not imply that generic quadrics from the second fundamental form are not degenerate. Indeed, the following theorem is just a reformulation of Theorem 6.1:

Theorem 6.29 If X is any projective variety and x ∈ X_{sm} then the projective second fundamental form |𝔽𝔽^{2}|_{x} of X at x is a system of quadrics of rank bounded above by dim X − def X. Moreover, if x ∈ X_{sm} is a generic point and Q ⊂ |𝔽𝔽^{2}|_{x} is a generic quadric then rank Q = dim X − def X.

6.4.B Moving Frames

The most convenient formalism for dealing with second (and higher) fundamental forms is the language of moving frames. We shall recall it briefly here, more details and applications can be found in [GH].

In ℙ^{N} a frame is denoted by {A_{0}, …, A_{N}}. It is given by a basis A_{0}, …, A_{N} for ℂ^{N+1}. The set of all frames forms an algebraic variety F(ℙ^{n}) the principal homogeneous space of the general linear group GL_{N+1}. Each of the vectors A_{i} may be viewed as a mapping v : F(ℙ^{N}) → ℂ^{N+1}. Expressing the exterior derivative dv in terms of the basis {A_{i}} gives

dA_{i} = ∑_{j} ω_{ij}A_{j}.

The (N +1)^{2} differential forms ω_{ij} are the Maurer–Cartan forms on the group GL_{N+1}, and by taking their exterior derivatives we obtain the Maurer–Cartan equations

dω_{ij} = ∑_{k} ω_{ik} ∧ ω_{kj}.

Geometrically we may think of a frame {A_{0}, …, A_{N}} as defining a coordinate simplex in ℙ^{N}, and then ω_{ij} gives the rotation matrix when this coordinate simplex is infinitesimally displaced.

There is a fibering π : F(ℙ^{N}) → ℙ^{N} given by

π{A_{0}, …, A_{N}} → A_{0}.

If we set ω_{i} = ω_{0i} then these 1-forms are horizontal for the fibering π, i.e., they vanish at the fibers π^{−1}(p), p ∈ ℙ^{N}. From dω_{i} = ∑_{j} ω_{j} ∧ ω_{ji} we see that the forms {ω_{i}} satisfy the Frobenius integrability condition. Thus we may think of the fibration π as defined by the foliation

ω_{1} = … = ω_{N} = 0.

The equation

dA_{0} = ∑_{i=0}^{N} ω_{i}A_{i} = ∑_{i=1}^{N} ω_{i}A_{i} mod A_{0}

has the following geometric interpretation. For each choice of the frame {A_{0}, …, A_{N}} lying over p ∈ ℙ^{N} the horizontal 1-forms ω_{1}, …, ω_{N} give a basis for the cotangent space T_{p}^{∗}ℙ^{N}. The corresponding basis v_{1}, …, v_{N} ∈ T_{p}ℙ^{N} for the tangent space has the property that v_{i} is tangent to the line \overline{A_{0}A_{i}}.

Assume now that we are given a connected n-dimensional smooth subvariety M ⊂ ℙ^{N}, not necessarily closed. Associated to M is the submanifold F(M) ⊂ F(ℙ^{N}) of Darboux frames

{A_{0}; A_{1}, …, A_{n}; A_{n+1}, …, A_{N}}

defined by the conditions that A_{0} lies over some p ∈ M and A_{0}, A_{1}, …, A_{n} span T̂_{p}M, the embedded tangent space of M at p. On F(M) we have the condition that differential 1-forms ω_{1}, …, ω_{n} give a basis of T_{p}^{∗}M and ω_{i} = 0 for i > n. Therefore it is natural to think of the ω_{1}, …, ω_{n} as homogeneous coordinates in the projectivized tangent spaces ℙ(T_{p}M). For example, a quadric in ℙ(T_{p}M) is defined by an equation ∑_{i,j=1}^{n} q_{ij}ω_{i}ω_{j} = 0, q_{ij} = q_{ji}.

Since ω_{i} = 0 for i > n, we also have dω_{i} = 0 on F(M). On the over hand, by the structure equations we have dω_{i} = ∑_{j=1}^{N} ω_{j} ∧ ω_{ji}. Therefore, for any i > n we have

∑_{j=1}^{n} ω_{j} ∧ ω_{ji} = 0.

Since ω_{1}, …, ω_{n} are linearly independent, an easy calculation (called Cartan Lemma) shows that for any μ > n, for any i = 1, …, n we have

ω_{iμ} = ∑_{j=1}^{n} q_{ijμ}ω_{j}, q_{ijμ} = q_{jiμ}.

We set for any μ = n + 1, …, N

Q_{μ} = ∑_{i,j=1}^{n} q_{ijμ}ω_{i}ω_{j}.

The equation Q_{μ} = 0 defines a quadric hypersurface in ℙ(T_{p}M).

Then an easy local calculation (see [GH1]) shows that we have

Proposition 6.30 The linear system of quadrics in ℙ(T_{p}M) spanned by Q_{μ}, μ = n + 1, …, N is the projective second fundamental form |𝔽𝔽^{2}|.

Thus, if we think of 𝔽𝔽^{2} as of the map S^{2}T_{p}M → (N_{M}ℙ^{N})_{p} and identify (N_{M}ℙ^{N})_{p} with ℂ^{N+1}/ Cone(T̂_{p}M) then for any vector v ∈ T_{p}M the vector 𝔽𝔽^{2}(v) is given in coordinates by

𝔽𝔽^{2}(v) = ∑_{i,j=1}^{n} ∑_{μ=n+1}^{N} ω_{i}(v)ω_{j}(v)A_{μ},

where v = ∑_{i=1}^{n} ω_{i}(v)v_{i} ∈ T_{p}M.

Another geometric interpretation of the second fundamental form is via the classical Meusnier–Euler Theorem.

Definition 6.31 If {p(t)} is any holomorphic arc in ℙ^{N} described by a vector–valued function A_{0}(t), then the osculating sequence is the sequence of linear spaces spanned by the following collection of vectors

{A_{0}(t)}, {A_{0}(t), A′_{0}(t)}, {A_{0}(t), A′_{0}(t), A′′_{0}(t)}, …

We then have the Meusnier–Euler Theorem

Theorem 6.32 ([GH1]) For a tangent vector v ∈ T_{p}M, the normal vector 𝔽𝔽^{2}(v, v) ∈ (N_{M}ℙ^{N})_{p} gives the projection in ℂ^{n+1}/ Cone(T̂_{p}M) of the second osculating space to any curve p(t), p(0) = p with tangent v at t = 0

Proof. We choose an arbitrary field of Darboux frames {A_{i}(t)} along p(t) and write

\frac{dA_{0}}{dt} = ∑_{i=1}^{n} (\frac{ω_{i}}{dt} ) A_{i} mod A_{0}.

Recall that ω_{i} is a 1-form on the frame manifold, so the expression \frac{ω_{i}}{dt} makes sense and is equal to the value of this 1-form on the tangent vector to the curve {A_{i}(t)} on the frame manifold. If we interpret {ω_{1}, …, ω_{n}} as a basis in the cotangent space T_{p}^{∗}M then this equation shows that \frac{ω_{i}}{dt} is equal to the i-th coordinate of v ∈ T_{p}M in the dual basis {v_{1}, …, v_{n}} of T_{p}M (in fact, v_{i} = A_{i} mod A_{0}).

Differentiating further, we get

\frac{d^{2}A_{0}}{dt^{2}} = ∑_{i=1}^{n} (\frac{ω_{i}}{dt} ) (\frac{dA_{i}}{dt} ) mod A_{0}, …, A_{n}

= ∑_{i,j=1}^{n} ∑_{μ=n+1}^{N} q_{ijμ} (\frac{ω_{i}}{dt} ) (\frac{ω_{j}}{dt} ) mod Cone(T̂_{p}M).

But we also have

𝔽𝔽^{2}(v, v) = ∑_{i,j=1}^{n} ∑_{μ=n+1}^{N} q_{ijμ} (\frac{ω_{i}}{dt} ) (\frac{ω_{j}}{dt} ) .

This finishes the proof. ⊓ ⊔

Finally, let us give a useful alternative coordinate description of the second fundamental form of M in ℙ^{N}. This one will be absolutely identical to the description implicitly used in the proofs of Theorems 6.1 and 6.12. For p ∈ M, we choose a homogeneous coordinate system X_{0}, …, X_{N} for ℙ^{N} so that p = (1 : 0 : …: 0) and T̂_{p}M is spanned by the first n + 1 coordinate vectors. If x_{1} = X_{1}/X_{0}, …, x_{N} = X_{N}/X_{0} is the corresponding affine coordinate system for the affine chart of ℙ^{N} and (z_{1}, …, z_{n}) is any holomorphic coordinate system for M centered at p, then for μ = n + 1, …, N we have in the neighborhood of p that x_{μ}|_{M} = x_{μ}(z) vanishes to second order at z = 0. Therefore,

x_{μ}(z) = ∑_{i,j=1}^{n} q_{ijμ}z_{i}z_{j} + (higher order terms).

The second fundamental form is the linear system of quadrics generated by ∑_{i,j=1}^{n} q_{ijν}dz_{i}dz_{j}.

6.4.C Fundamental Forms of Projective Homogeneous Spaces

It is interesting to explicitly compute the second fundamental form of some familiar homogeneous spaces.

Example 6.33 ([GH1]) Suppose that ℙ^{m}×ℙ^{n} ⊂ ℙ^{nm+n+m} is the Segre embedding. Recall that if ℙ^{m} = ℙ(V), ℙ^{n} = ℙ(W) then the image M of the Segre variety is the image of the natural inclusion ℙ(V)×ℙ(W) ⊂ ℙ(V ⊗W). Choosing coordinates {X_{α}} in V and {Y_{μ}} in W, this inclusion is given by mapping ({X_{α}}, {Y_{μ}}) → {X_{α}Y_{μ}}. We denote by {A_{0}, …, A_{m}} and {B_{0}, …, B_{n}} frames for ℙ(V) and ℙ(W). We shall use the range of indices 1 ≤ i, j ≤ m, 1 ≤ α, β ≤ n. If A_{0} lies over v_{0} ∈ ℙ(V) and B_{0} over w_{0} ∈ ℙ(W), then the frame

{A_{0} ⊗ B_{0}, A_{0} ⊗ B_{α}, A_{i} ⊗ B_{0}, A_{i} ⊗ B_{α}}  (6.2)

in ℙ(V ⊗ W) lies over v_{0} ⊗ w_{0}. If we write

dA_{0} = ∑_{i} ϕ_{i}A_{i} mod A_{0}, dB_{0} = ∑_{α} ψ_{α}B_{α} mod B_{0},

then

d(A_{0} ⊗ B_{0}) = ∑_{α} ψ_{α}A_{0} ⊗ B_{α} + ∑_{i} ϕ_{i}A_{i} ⊗ B_{0} mod A_{0} ⊗ B_{0}.

Therefore, (6.2) is a Darboux frame for M and

T̂_{v_{0}⊗w_{0}}M = ℙ(V ⊗ w_{0} + v_{0} ⊗ W),

so that

(N_{M})_{v_{0}⊗w_{0}} = (V ⊗ W)/(V ⊗ w_{0} + v_{0} ⊗ W).

Differentiating further, we get

d^{2}(A_{0} ⊗ B_{0}) = 2 ∑_{i,α} ϕ_{i}ψ_{α}A_{i} ⊗ B_{α} mod {A_{0} ⊗ B_{0}, A_{i} ⊗ B_{0}, A_{0} ⊗ B_{α}}.

Therefore, the second fundamental form is given by

𝔽𝔽^{2}(v_{0} ⊗ w + v ⊗ w_{0}, v_{0} ⊗ w + v ⊗ w_{0}) = 2v ⊗ w mod V ⊗ w_{0} + v_{0} ⊗ W.

In terms of homogeneous coordinates {ϕ_{i}} for T_{v_{0}}ℙ^{m} and {ψ_{α}} for T_{w_{0}}ℙ^{n}, |𝔽𝔽^{2}| is the linear system of quadrics

∑_{i,α} q_{iα}ϕ_{i}ψ_{α}

for any matrix q_{iα}. In other words, matrices of quadrics from this linear system have the form

( \begin{matrix}0 & Q \\ Q^{t} & 0 \end{matrix} ) ,

where Q is an arbitrary matrix.

Summarizing, we see that the projectivization of the tangent space at any point x ∈ M is ℙ^{n+m−1} and |𝔽𝔽^{2}|_{x} is the complete linear system of quadrics having as a base locus the union ℙ^{m−1} ∩ ℙ^{n−1} of two skew linear subspaces.

Example 6.34 ([GH1]) Suppose that ℙ^{n} = ℙ(V) ⊂ ℙ(S^{d}V) is the d-th Veronese embedding. In terms of homogeneous coordinates {X_{0}, …, X_{n}} for ℙ(V) the embedding is given by

(…: X_{α} : …) ↦ (X_{0}^{d} : X_{0}^{d−1}X_{1} : …: X_{0}^{d−1}X_{n} : …: F_{λ}(X) : …),

where F_{λ} varies over a basis for the homogeneous forms of degree d. Given a point v_{0} ∈ ℙ^{n}, e.g., v_{0} = (1 : …: 0), v_{0} maps to (1 : …: 0); the projectivization of the tangent space at v_{0} is ℙ^{n−1} spanned by first n coordinate vectors in ℙ(S^{d}V). The affine coordinate system in the affine chart containing v_{0} is given by F_{λ}(X)/X_{0}^{d} = F_{λ}(X_{1}/X_{0}, …, X_{n}/X_{0}). Therefore, |𝔽𝔽^{2}|_{v_{0}} is the linear system of all quadrics.

Example 6.35 ([GH1]) Let Gr(k, V) ⊂ ℙ(Λ^{k}V) be the Grassmanian in the Plücker embedding, dim V = n. Then the projectivization of the tangent space at any point x is ℙ(Mat_{k,n−k}) and an easy calculation (see [GH1]) shows that |𝔽𝔽^{2}|_{x} generated by (2 × 2)-minors of this matrix. Using this description, it is quite straightforward to find the defect of the Grassmanian in the Plücker embedding. All we need to do is to find the maximal possible rank of a quadric generated by (2 × 2)-minors of a rectangular matrix. The answer is as follows. We may suppose that k ≤ n/2. Then the dual variety is always a hypersurface except the case k = 1 (when the dual variety is empty) and the case k = 2, n is odd (when the defect is equal to 2).

Example 6.36 ([Lan4]) Let 𝕊_{m} ⊂ ℙ(Λ^{ev}ℂ^{m}) be the spinor variety (c.f. Example 2.2). Then calculating the tangent space at the point v_{0} = 1 it is easy to see that T_{v_{0}}𝕊_{m} = Λ^{2}ℂ^{m}. Let Gr(2, ℂ^{m}) ⊂ ℙ(Λ^{2}ℂ^{m}) be the Plücker embedding then the linear system |𝔽𝔽^{2}| consists of all quadrics containing Gr(2, ℂ^{m}).

Example 6.37 ([Lan4]) Consider the Severi variety, c.f. Example 2.3. Let 𝔻_{ℝ} denote one of four division algebras over ℝ: real numbers ℝ, complex numbers ℂ, quaternions ℍ, or octonions 𝕆. Let 𝔻 = 𝔻_{ℝ} ⊗_{ℝ} ℂ denotes its complexification, therefore 𝔻 is either ℂ, ℂ ⊕ ℂ, Mat_{2}(ℂ), or ℂa (the algebra of Cayley numbers). All these algebras have the standard involution. Let H_{3}(𝔻) denote the 3 × 3 Hermitian matrices over 𝔻. If x ∈ H_{3}(𝔻) then we may write

x = ( \begin{matrix}c_{1} & u_{1} & u_{2} \\ \overline{u_{1}} & c_{2} & u_{3} \\ \overline{u_{2}} & \overline{u_{3}} & c_{3} \end{matrix} ) , c_{i} ∈ ℂ, u_{i} ∈ 𝔻.

The projectivization of the highest weight vector orbit with respect to the group of norm similarities is called the Severi variety corresponding to D. One may think of Severi varieties as of complexifications of projective planes over 𝔻_{ℝ}. Take a point p of the Severi variety of the form

p = ( \begin{matrix}1 & 0 & 0 \\ 0 & 0 & 0 \\ 0 & 0 & 0 \end{matrix} ) .

Choose the affine coordinates u_{1}, u_{2}, u_{3} ∈ 𝔻, c_{2}, c_{3} ∈ ℂ of H_{3}(𝔻) centered at p. Then the tangent space at p is {u_{1}, u_{2}} and the second fundamental form is generated by quadrics

u_{1}\overline{u_{1}} = 0, u_{2}\overline{u_{2}} = 0, \overline{u_{2}}u_{1} = 0.

For example, if 𝔻 = ℂa then the base locus of this linear system is the spinor variety 𝕊_{5} (c.f. Example 2.21).

The higher fundamental forms are defined similar to the second fundamental form 𝔽𝔽^{2}. The most geometric way is to utilize the Euler–Meusnier Theorem 6.32 as the definition.

Recall that if {p(t)} is any holomorphic arc in ℙ^{N} described by a vector– valued function A_{0}(t), then the osculating sequence at p = p(0) is the sequence of linear spaces spanned by the following collection of vectors

{A_{0}(t)}, {A_{0}(t), A′_{0}(t)}, {A_{0}(t), A′_{0}(t), A′′_{0}(t)}, …

Suppose now that M ⊂ ℙ^{N} = ℙ(V) is a smooth variety, p ∈ M, T̃_{p}M ⊂ V is the cone over the embedded tangent space T̂_{p}M. Then the sequence of osculated spaces T_{p}^{(0)}M = ℂp, T_{p}^{(1)}M = T̃_{p}M, T_{p}^{(2)}M, … is defined as follows. The osculating space T_{p}^{(k)}M is the linear span of k-th osculating spaces for smooth curves p(t) ⊂ M, p(0) = p. The k-th normal space N_{p}^{(k)}M is defined as V/T_{p}^{(k)}M, so we have N_{p}^{(1)}M = (N_{M}ℙ^{N})_{p}. Suppose that v ∈ T_{p}M = T_{p}^{(1)}M/T_{p}^{(0)}M is any tangent vector, {p(t)} is any holomorphic arc in ℙ^{N} with tangent vector v at t = 0. Let A(t) ⊂ V be a curve projected to p(t). Then it is easy to see that the vector

𝔽𝔽^{k}(v) = \frac{d^{k}A(t)}{dt^{k}} mod T_{p}^{(k−1)}M ∈ N_{p}^{(k−1)}M

depends only on v. This construction gives a map

𝔽𝔽^{k} : S^{k}T_{p}M → N_{p}^{(k−1)}M

called the k-th projective fundamental form. Of course, it can be also described via moving frames, in coordinates, by differentiating Gauss map, etc., see e.g. [GH1, Lan4]. The sequence of dimensions of osculating spaces T_{p}^{(k)}M can, of course, depend on p ∈ M. This will not be very essential, first, because we can restrict ourselves to an open subset of M where these dimensions are fixed. Secondly, we shall be interested only in projective homogeneous varieties in equivariant embeddings, in which case all points have the same rights.

Example 6.38 It is easy to see that if M ⊂ ℙ^{N} is a hypersurface of degree d, p ∈ M is a smooth point, and l ⊂ ℙ(T_{p}^{(d)}M) is a line passing through p then l ⊂ M. More generally, if M ⊂ ℙ^{N} is a projective variety whose homogeneous ideal is generated by elements of degree at most d, p ∈ M is a smooth point, and l ⊂ ℙ(T_{p}^{(d)}M) is a line passing through p then l ⊂ M. If V is an irreducible module of a simple group G and M = G/P ⊂ ℙ(V) is the projectivization of the orbit of the highest weight then its ideal is generated in degree two (see e.g. [Li]). In particular, for any p ∈ M Base |𝔽𝔽^{2}|_{p} is the set of tangent directions to lines on M through p. Using this idea, in [LM] the thorough study of varieties of lines and linear subspaces on minimal equivariant projective embeddings of minimal flag varieties was undertaken.

Let G be a connected semisimple complex Lie group, 𝔤 its Lie algebra, U𝔤 the universal enveloping algebra of 𝔤. U𝔤 has a natural filtration U𝔤^{(k)} such that the associated graded algebra is the symmetric algebra S𝔤 of 𝔤.

Let V_{λ} be an irreducible G-module with highest weight λ, and v_{λ} ∈ V_{λ} a highest weight vector. The action of U𝔤 on V_{λ} induces a filtration of V_{λ} whose k-th term is V_{λ}^{(k)} = U𝔤^{(k)}v_{λ}.

Let x = [v_{λ}] ∈ ℙ(V_{λ}) be the line in V_{λ} generated by v_{λ}, and let X = G/P ⊂ ℙ(V_{λ}) be the projectivization of the orbit of the highest weight vector. Here P ⊂ G is a parabolic subgroup, namely, the stabilizer of x. Let 𝔭 be the Lie algebra of P. The tangent bundle T X is a homogeneous vector bundle and we can canonically identify T_{x}X with 𝔤/𝔭 as a P -module. Then it is easy to verify that the osculating spaces and the fundamental forms of X have a following simple representation-theoretic interpretation.

Proposition 6.39 Let X = G/P ⊂ ℙ(V_{λ}) be a polarized flag variety and let x = eP ∈ X. Then the k-th osculating space T_{x}^{(k)}X = V_{λ}^{(k)}, therefore N_{x}^{(k)}X = V_{λ}/V_{λ}^{(k)}. Moreover, there is a commutative diagram

S^{k}𝔤  U𝔤^{(k)}/U𝔤^{(k−1)}
π_{1}↓  ↓^{π_{2}}
S^{k}(T_{x}X) =S^{k}(𝔤/𝔭) ⟶_{𝔽𝔽^{k}} V_{λ}/V_{λ}^{(k)} = N_{x}^{(k−1)}X

where π_{1} is induced by the projection 𝔤 → 𝔤/𝔭 and π_{2} is the map given by u ↦ u · v_{λ}.

7. The Degree of a Dual Variety

Preliminaries

Let X ⊂ ℙ(V) be a smooth projective variety, L = O_{X}(1). In this chapter we review several approaches for finding the degree of the dual variety X^{∗}. In Section 7.1 we give several applications of the Katz–Kleiman Holme formula. In Section 7.2 we collect various calculations of degrees of discriminants of polarized flag varieties. In final sections we study the degree of the discriminant of a polarized variety as a function at polarization and express the degree in terms of Hilbert polynomials.

7.1 Katz–Kleiman–Holme Formula

For any vector bundle E on X we denote by c_{i}(E) its i-th Chern class (see e.g. [Fu1]). For example, c_{1}(L) = H, the hyperplane section divisor of X in ℙ(V). For any 0-dimensional cycle Z on X we denote its degree (‘the number of points’ in Z) by ∫_{X} Z. Let J(L) be the first jet bundle of L. Let dim X = n.

Theorem 7.1 ([BFS1])

(a) deg Δ_{X} = ∫_{X} c_{n}(J(L)). In particular, X^{∗} is a hypersurface if and only if c_{n}(J(L)) ≠ 0.

(b) Moreover, def X = k if and only if c_{r}(J(L)) = 0 for r ≥ n − k + 1, and c_{n−k}(J(L)) ≠ 0. In this case

deg X^{∗} = ∫_{X} c_{n−k}(J(L)) · H^{k}.

Sketch of the proof. Suppose that def X = k. Notice that jets j(f), f ∈ V^{∗}, span J(L). We choose generic elements f_{1}, …, f_{v} ∈ V^{∗}, where v = dim V^{∗}. Using the classical representation of a Chern class of a spanned vector bundle (see e.g. [Fu1]), we have that c_{n−k+1}(J(L)) is represented by the set

{x ∈ X | j(f_{1})(x), …, j(f_{k+1})(x) are linearly dependent}.

Since codim_{ℙ(V^{∗})} X^{∗} = k + 1, generic k-dimensional projective subspace in ℙ(V^{∗}) does not meet X^{∗}. Therefore, any non-trivial linear combination of f_{1}, …, f_{k+1} does not belong to the Cone(X^{∗}). But this is equivalent to

dim Span{j(f_{1})(x), …, j(f_{k+1})(x)} = k + 1

for any x ∈ X. This implies by the above representation that c_{n−k+1}(J(L) = 0 and moreover that c_{r}(J(L)) = 0 for r ≥ n − k + 2. Now, the Chern class c_{n−k}(J(L)) is represented by the set

{x ∈ X | j(f_{1})(x), …, j(f_{k+2})(x) are linearly dependent}.

This set is a union of the singular loci of the hyperplane sections H ∩ X, where H ∈ X^{∗} ∩ L for a generic (k + 1)-dimensional subspace L ⊂ ℙ(V^{∗}). Notice that the number of these hyperplane sections is equal to the degree of the dual variety deg X^{∗}. Moreover, by dimension reasons we may suppose that X^{∗} ∩ L ⊂ X^{∗}_{sm}. By Theorem 1.12 these singular loci are projective subspaces of dimension k. Therefore c_{n−k}(J(L)) is represented by deg X^{∗} k-dimensional projective subspaces. In particular, c_{n−k}(J(L)) ≠ 0 and deg X^{∗} = ∫_{X} c_{n−k}(J(L)) · H^{k}. ⊓ ⊔

Using the exact sequence (3.1) it is possible to substitute Chern classes of the jet bundle by Chern classes of the cotangent bundle Ω_{X}^{1} = T_{X}^{∗}. The resulting formula was found by N. Katz and S. Kleiman [Ka, Kl1] in the case where X^{∗} is a hypersurface and by A. Holme [Ho1, Ho2] in the general case. Consider the Chern polynomial of X with respect to the given projective embedding

c_{X}(q) = ∑_{i=0}^{n} q^{i+1} ∫_{X} c_{n−i}(Ω_{X}^{1}) · H^{i}.

Theorem 7.2

(a) deg Δ_{X} = c′_{X}(1) = ∑_{i=0}^{n}(i + 1) ∫_{X} c_{n−i}(Ω_{X}^{1}) · H^{i}.

(b) The codimension of X^{∗} equals the order of the zero at q = 1 of the polynomial c_{X}(q) − c_{X}(1). If this order is μ then deg X^{∗} = c_{X}^{(μ)}(1)/μ!.

The formula in (a) can be rewritten in a numerous number of ways, sometimes more convenient for calculations. The proofs of the following formulas can be found in [Kl1].

Example 7.3 ([Kl1])

deg Δ_{X} = ∫_{X} s_{n}(N_{X}ℙ^{N}(−1)),

where s_{n} is the n-th Segre class (see e.g. [Fu1]).

Example 7.4 ([Kl1])

deg Δ_{X} = ∫_{X} \frac{1}{c(N_{X}^{∗}ℙ^{N}(1))}.

For example, if X is a hypersurface of degree d in ℙ^{N}, then N_{X}^{∗}ℙ^{N}(1) = O_{X}(1 − d) and

deg Δ_{X} = ∫_{X} \frac{1}{1 − (d − 1)H} = ∫_{X}(d − 1)^{n}H^{n} = d(d − 1)^{n}.

In particular, X^{∗} is a hypersurface (if X is not a hyperplane). In the same way one can show that X^{∗} is a hypersurface if X is a non-degenerate smooth complete intersection (c.f. Theorem 4.2).

Example 7.5 ([Kl1, Lam]) The following non-trivial consequence of Theorem 7.2 is called the class formula.

deg Δ_{X} = (−1)^{n} [χ(X) − 2χ(X ∩ H) + χ(X ∩ H ∩ H′)] ,

where H and H′ are generic hyperplanes (such that X ∩H and X ∩H ∩H′ are smooth) and χ is a topological Euler characteristic. This formula is the main ingredient in the proof of the following Landman formula (see [Lan, Kl3]):

deg Δ_{X} = (b_{n}(X) − b_{n−2}(X)) + 2(b_{n−1}(X ∩ H) − b_{n−1}(X))
+(b_{n−2}(X ∩ H ∩ H′) − b_{n−2}(X)),

where b_{i} is the i-th Betti number. All three summands are nonnegative numbers due to the strong and weak Lefschetz theorems. Hence the characteristic condition for the positive defect is

b_{n}(X) = b_{n−2}(X), b_{n−1}(X ∩ H) = b_{n−1}(X), b_{n−2}(X ∩ H ∩ H′) = b_{n−2}(X).

Example 7.6 ([Kl1]) Let X ⊂ ℙ^{2} be a smooth plane curve of degree d. Consider the Veronese embedding ℙ^{2} ⊂ ℙ^{N}, N + 1 = \binom{r+2}{2}, and the corresponding embedding X ⊂ ℙ^{N}. Then the dual variety X^{∗} ⊂ ℙ^{N}^{∗} parametrizes plane curves of degree r tangent to X at some point. It is easy to see that

deg Δ_{X} = d(d + 2r − 3).

For example, the variety of conics tangent to a given conic is a hypersurface in ℙ^{5} of degree 6.

Example 7.7 Suppose that X ⊂ ℙ^{N} is a smooth projective curve of genus g and degree d. Let H ⊂ X be a hyperplane section. Then X^{∗} is a hypersurface by Example 1.13, and its degree is given by

deg Δ_{X} = deg K_{X} + 2 deg H = 2g − 2 + 2d,

where K_{X} denotes the canonical divisor.

Example 7.8 Let X ⊂ ℙ^{N} be a smooth projective surface, H ⊂ X be a smooth hyperplane section. Then X^{∗} is a hypersurface by Example 6.16, and its degree is given by

deg Δ_{X} = deg c_{2}(Ω_{X}^{1}) + 2K_{X} · H + 3H · H = χ(X) − 2χ(H) + H · H,

where χ is the topological Euler characteristic.

Example 7.9 Suppose that X = ℙ^{n} = ℙ(ℂ^{n+1}) is embedded into ℙ(S^{d}ℂ^{n+1}) via the Veronese embedding. Then elements of X^{∗} parametrize hypersurfaces with singularities of degree d in ℙ^{n}, therefore Δ_{X} is the classical discriminant of homogeneous forms of degree d in n + 1 variables. Let us find its degree. The Chow ring (or the cohomology ring) of X is equal to ℤ(t)/t^{n+1}, the element t being the class of the hyperplane. The first Chern class of the line bundle corresponding to the Veronese embedding is equal to dt. The total Chern class ∑_{i} c_{i}(Ω_{ℙ^{n}}^{1}) equals (1 − t)^{n+1}, therefore c_{i}(Ω_{ℙ^{n}}^{1}) = (−1)^{i}\binom{n+1}{i}t^{i}. Clearly ∫_{X} t^{n} = 1. So we have

c_{X}(q) = ∑_{i=0}^{n} q^{i+1}(−1)^{n−i}\binom{n + 1}{i + 1}d^{i} = \frac{(qd − 1)^{n+1} − (−1)^{n+1}}{d} .

Therefore, deg Δ_{X} = c′_{X}(1) = (n + 1)(d − 1)^{n}. This formula is known as a Boole formula.

A little bit more explicit form of Theorem 7.2 can be given as follows.

Definition 7.10 Let X be a n-dimensional smooth projective variety in ℙ^{N}, L = O_{X}(1). Then the s-th rank of X with respect to the projective embedding is defined as

δ_{s} = ∑_{i=s}^{n} \binom{i + 1}{s + 1}e_{n−i},

where e_{j} denotes the degree of c_{j}(T^{∗}X) with respect to the given projective embedding.

Then Theorem 7.2 can be reformulated as follows

Corollary 7.11 The defect def X is the smallest integer r ≥ 0 such that δ_{r} ≠ 0, and for r = def X the degree of the dual variety X^{∗} equals δ_{r}.

7.2 Degrees of Discriminants of Polarized Flag Varieties

7.2.A The Degree of the Dual Variety to G/B

We follow the notation of Section 2.1. Let G be a simple algebraic group of rank r with Borel subgroup B. The minimal equivariant projective embedding of G/B corresponds to the line bundle L_{ρ}, where ρ = ω_{1} + … + ω_{r} is a half sum of positive roots. We shall follow [CW] here and calculate the degree of the corresponding discriminant (in fact, in most cases the dual variety will be a hypersurface, see precise statements in Section 9.3).

Let {β_{1}, …, β_{N}} = Δ_{+} be the collection of all positive roots and let {β_{1}^{∨}, …, β_{N}^{∨}} = Δ_{+}^{∨} be the collection of all positive coroots. We consider the matrix

M = (\frac{(β_{i}^{∨}, β_{j})}{(β_{i}^{∨}, ρ)})_{i,j=1,...,N} .

Let P_{s}(M) be the sum of all permanents of s × s submatrices of M. Recall

that the permanent of a n × n matrix (a_{ij}) is equal to ∑_{σ∈S_{n}} ∏_{i=1}^{n} a_{i,σ(i)}.

Theorem 7.12 ([CW])

deg Δ_{(G/B,L_{ρ})} = ∑_{s=0}^{N}(s + 1)!P_{N−s}(M).

Proof. Recall that by the Kleiman formula 7.2 we have

deg Δ_{(G/B,L_{ρ})} = ∑_{i=0}^{N}(i + 1) ∫_{G/B} c_{n−i}(Ω_{G/B}^{1}) · c_{1}(L_{ρ})^{i}.

Therefore we need to calculate these integrals. The correspondence λ ↦ L_{λ} identifies P_{ℚ} (the character lattice tensored by ℚ) with Pic(G/B)⊗ℚ. Therefore, we have a homomorphism of commutative algebras

c : S^{•}(P_{ℚ}) → A^{•}(G/B),

where S^{•}(P_{ℚ}) is the symmetric algebra of P_{ℚ} and A^{•}(G/B) is the rational Chow ring. It is well-known (see e.g. [BGG]) that this homomorphism is surjective and its kernel is generated as an ideal by W -invariants (S^{•}(P_{ℚ}))_{+}^{W} of positive degree, where W is the Weyl group. For example,

c_{1}(L_{ρ}) = c(ρ).

Using an appropriate filtration of the cotangent bundle Ω_{G/B}^{1} and ‘the splitting principle’ it is easy to see that

c_{n−i}(Ω_{G/B}^{1}) = c(X_{i}), X_{i} = ∑_{\begin{matrix}Γ ⊂Δ_{+} \\ #Γ =n−i \end{matrix}} ∏_{α∈Γ} α.

Therefore, it is sufficient to compute c(ρ^{i}X_{i}).

For any coroot α^{∨} ∈ P_{ℚ}^{∗} we define a linear function

D_{α^{∨}}(λ) = ⟨α^{∨}, λ⟩.

This linear function extends by a Leibniz rule to the differential operator on S^{•}P_{ℚ}. We consider now the differential operator D on S^{•}P_{ℚ} given by

D = \frac{∏_{α>0} D_{α^{∨}}}{∏_{α>0}(α^{∨}, ρ)}.

The operator D decreases the degree by N, so in particular we get a linear form

D : S^{N}P_{ℚ} → ℚ.

One can show (see [CW]) that for any x ∈ S^{N}P_{ℚ} we have

D(x) = ∫_{G/B} c(x).

Now the claim of the theorem follows by an easy calculation. ⊓ ⊔

7.2.B Degrees of Hyperdeterminants

Consider the flag variety ℙ^{k_{1}} × … × ℙ^{k_{r}} of the group SL_{k_{1}} × … × SL_{k_{r}}. The projectively dual variety of its ‘minimal’ equivariant projective embedding corresponds to a nice theory of hyperdeterminants that was initiated by Cayley and Schläffli [Ca1, Ca2, Schl].

Let r ≥ 2 be an integer, and A = (a_{i_{1}...i_{r}}), 0 ≤ i_{j} ≤ k_{j} be an r-dimensional complex matrix of format (k_{1}+1)×…×(k_{r}+1). The hyperdeterminant of A is defined as follows, c.f. 6.2.B. Consider the product X = ℙ^{k_{1}} ×…×ℙ^{k_{r}} of several projective spaces embedded into the projective space ℙ^{(k_{1}+1)×...×(k_{r}+1)−1} via the Segre embedding. Let X^{∗} be the projectively dual variety. If X^{∗} is a hypersurface then it is defined by a corresponding discriminant Δ_{X}, which in this case is called the hyperdeterminant (of format (k_{1} + 1) × … × (k_{r} + 1)), denoted by Det. Clearly Det(A) is a polynomial function in matrix entries of A invariant under the action of he group SL_{k_{1}+1} × …×SL_{k_{r}+1}. If X^{∗} is not a hypersurface then we set Det = 1. Theorem 6.8 shows that X^{∗} is a hypersurface (and, hence, defines a hyperdeterminant) if and only if 2k_{j} ≤ k_{1}+…+k_{r} for j = 1, …, r. If for one of j we have an equality 2k_{j} = k_{1} + … + k_{r} then the format is called boundary. Let N(k_{1}, …, k_{r}) be the degree of the hyperdeterminant of format (k_{1} + 1) × … × (k_{r} + 1). The proof of the following theorem can be found in [GKZ2] or [GKZ3].

Theorem 7.13

(a) The generating function for the degrees N(k_{1}, …, k_{r}) is given by

∑_{k_{1},...,k_{r}≥0} N(k_{1}, …, k_{r})z_{1}^{k_{1}} … z_{r}^{k_{r}} = \frac{1}{(1 − ∑_{i=2}^{r}(i − 1)e_{i}(z_{1}, …, z_{r}))^{2}} ,

where e_{i}(z_{1}, …, z_{r}) is the i-th elementary symmetric polynomial.

(b) The degree N(k_{1}, …, k_{r}) of the boundary format is (assuming that k_{1} = k_{2} + … + k_{r})

N(k_{2} + … + k_{r}, k_{2}, …, k_{r}) = \frac{(k_{2} + … + k_{r} + 1)!}{k_{2}! … k_{r}!}.

(c) The degree of the hyperdeterminant of the cubic format is given by

N(k, k, k) = ∑_{0≤j≤k/2} \frac{(j + k + 1)!}{(j!)^{3}(k − 2j)!} · 2^{k−2j}.

(d) The exponential generating function for the degree N_{r} of the hyperdeterminant of format 2 × 2 × … × 2 (r times) is given by

∑_{r≥0} N_{r} \frac{z^{r}}{r!} = \frac{e^{−2z}}{(1 − z)^{2}} .

7.2.C One Calculation

In most circumstances, known formulas for the degree of the discriminant depend on the certain set of discrete parameters. There are two possibilities for these parameters. First, we may fix a projective variety and vary its polarizations. The degree of the discriminant as a function in polarization is studied in the next section. Second possibility is to change a variety and to fix a polarization (in a certain sense). For example, consider the irreducible representation of SL_{n_{0}} with the highest weight λ Then this weight can be considered as a highest weight of SL_{n} for any n ≥ n_{0} with respect to the natural embedding SL_{n} ⊂ SL_{n+1} ⊂ . . .. As a result, we shall obtain a tower of flag varieties with “the same” polarization. The degree of the corresponding discriminants will be a function in n. This function can be very complicated. For example, in [Las] a very involved combinatorial formula for this degree was found in the case λ is a fundamental weight (i.e. for the degree of a dual variety of the Grassmanian Gr(k, n) in the Plücker embedding). However, sometimes this function has a closed expression. For example, the Boole formula 7.9 is a formula of this kind. Another simple example is the formula 2.20 for the degree of the dual variety to Gr(2, n) in the Plücker embedding. The following theorem is the mixture of these two cases. We are not aware of any other similar formulas.

Theorem 7.14 ([T1]) Let V be an irreducible SL_{n}-module with the highest weight (a − 1)φ_{1} + φ_{2}, a ≥ 2. Then the variety X^{∗} ⊂ ℙ(V^{∗}) projectively dual to the projectivization X of the orbit of the highest vector is a hypersurface of degree

\frac{(n^{2} − n)a^{n+1} − (n^{2} + n)a^{n−1} − 2n(−1)^{n}}{(a + 1)^{2}} .

Proof. We have to calculate the degree of Δ_{X}. We use Kleiman’s formula 7.2 for the degree of the dual variety. In the present case X = G/P, where G = SL_{n} and P ⊂ G is the parabolic subgroup of matrices

( \begin{matrix}∗ & 0 & 0 & … & 0 \\ ∗ & ∗ & 0 & … & 0 \\ ∗ & ∗ & ∗ & … & ∗ \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ ∗ & ∗ & ∗ & … & ∗ \end{matrix} ) .  (7.1)

Assume that T ⊂ G is the diagonal torus, B is the Borel subgroup of lowertriangular matrices, x_{1}, …, x_{n} are the weights of the tautological representation, X(T) is the lattice of characters of T, S is the symmetric algebra of X(T) (over ℚ), W ≃ S_{n} is the Weil group of G, and W_{P} ≃ S_{n−2} is the Weil group of P. It is well known (see [BGG]) that the map c : X(T) → Pic(G/B) that assigns to λ the first Chern class of the invertible sheaf L_{λ} can be extended to a surjective homomorphism c : S → A^{∗}(G/B) in the (rational) Chow ring, and its kernel coincides with S_{+}^{W} S. The projection α : G/B → G/P induces an embedding α^{∗} : A^{∗}(G/P) → A^{∗}(G/B). The image coincides with the subalgebra of W_{P} -invariants. Hence, A^{∗}(G/P) = S^{W_{P}} /S_{+}^{W} S^{W_{P}}. We denote the homomorphism S^{W_{P}} → A^{∗}(G/P) by the same letter c.

To apply Kleiman’s formula we need c_{1}(L) (which is equal to c(ax_{1} +x_{2})) and the total Chern class of Ω_{Z}^{1}, which is equal to

c(Ω_{Z}^{1}) = c ((1 − x_{1} + x_{2}) ∏_{i=3,...,n}(1 − x_{1} + x_{i}) ∏_{i=3,...,n}(1 − x_{2} + x_{i})) .

(This can be shown by standard arguments using the filtration of Ω_{Z}^{1} and splitting principle.)

Let α_{1}, …, α_{n−2} be the elementary symmetric polynomials in x_{3}, …, x_{n}. Then S^{W_{P}} = ℚ[x_{1}, x_{2}, α_{1}, …, α_{n−2}], and the ideal S_{+}^{W} S^{W_{P}} is generated by

x_{1} + x_{2} + α_{1}, x_{1}x_{2} + x_{1}α_{1} + x_{2}α_{1} + α_{2},

x_{1}x_{2}α_{1} + x_{1}α_{2} + x_{2}α_{2} + α_{3}, …, x_{1}x_{2}α_{n−4} + x_{1}α_{n−3} + x_{2}α_{n−3} + α_{n−2}, x_{1}x_{2}α_{n−3} + x_{1}α_{n−2} + x_{2}α_{n−2}, x_{1}x_{2}α_{n−2}.

Hence, α_{i} = (−1)^{i}(x_{1}^{i} + x_{1}^{i−1}x_{2} + … + x_{2}^{i}) mod S_{+}^{W} S^{W_{P}}, and A^{∗}(G/P) is isomorphic to the quotient ring ℚ[x_{1}, x_{2}]/⟨f_{1}, f_{2}⟩, where f_{1} = x_{1}^{n−1}+x_{1}^{n−2}x_{2}+ … + x_{2}^{n−1} and f_{2} = x_{1}^{n}. Note that f_{1}, f_{2} is the Gröbner basis of the ideal ⟨f_{1}, f_{2}⟩ with respect to the ordering x_{2} > x_{1} (see [Berg]). Hence, the set of X^{i}Y^{j}, where X = x_{1} mod ⟨f_{1}, f_{2}⟩, Y = x_{2} mod ⟨f_{1}, f_{2}⟩, i = 1, …, n − 1, j = 1, …, n − 2, is a basis of the quotient algebra.

To calculate the degree of the discriminant by Kleiman’s formula, we have to calculate ∫_{Z} c(X^{n−1}Y^{n−2}). Let w̃_{0} be the longest element in W, and let w_{0} be the shortest element in w̃_{0}W_{P} with the reduced factorization

w_{0} = ( \begin{matrix}1 & 2 & 3 & 4 & … & n \\ n & n − 1 & 1 & 2 & … & n − 2 \end{matrix} )

= (n − 1, n)(n − 2, n − 1) … (12) · (n − 1, n)(n − 2, n − 1) … (23).

Let

A_{w_{0}} = A_{(n−1,n)}A_{(n−2,n−1)} … A_{(12)}A_{(n−1,n)}A_{(n−2,n−1)} … A_{(23)}

be the corresponding endomorphism of degree −(2n − 3) in S, where

A_{(ij)} = \frac{id − s_{(ij)}}{x_{i} − x_{j}} ,

and s_{(ij)} is the reflection that transposes x_{i} and x_{j}. Then

∫_{Z} c(X^{n−1}Y^{n−2}) = A_{w_{0}}(x_{1}^{n−1}x_{2}^{n−2})

(see [BGG]). It is obvious that

A_{w_{0}}(x_{1}^{n−1}x_{2}^{n−2}) = A_{ρ_{1}}A_{ρ_{2}}(x_{1}^{n−1}x_{2}^{n−2})
= A_{ρ_{1}}(x_{1}^{n−1}A_{ρ_{2}}(x_{2}^{n−2})) = A_{ρ_{1}}(x_{1})A_{ρ_{2}}(x_{2}),

where ρ_{k} = (n − 1, n)(n − 2, n − 1) … (k, k + 1). These factors are both equal to 1, since they are equal to ∫_{ℙ^{n−1}} c_{1}(O(1))^{n−1} and ∫_{ℙ^{n−2}} c_{1}(O(1))^{n−2}, respectively, and it is obvious that these integrals are equal to 1. We finally obtain that ∫_{Z} c(X^{n−1}Y^{n−2}) = 1.

It remains to calculate the polynomial

∑_{i=0}^{2n−3}(i + 1)c_{2n−3−i}(aX + Y)^{i},  (7.2)

in the ring ℚ[X, Y] (with the basis X^{i}Y^{j}, i = 1, …, n−1, j = 1, …, n−2, and relations X^{n−1}+X^{n−2}Y +…+Y^{n−1} = 0, Y^{n} = 0, and X^{n} = 0, which follows from the preceding relations), where c_{k} is the kth homogeneous component of the polynomial

(1 − X + Y) ∏_{i=3}^{n}(1 − X + x_{i}) ∏_{i=3}^{n}(1 − Y + x_{i}),

in which the ith symmetric function of x_{3}, …, x_{n} must be replaced by (−1)^{i}(X^{i} + X^{i−1}Y + … + Y^{i}). By the previous discussion, the result of this calculation will be deg(Δ_{X})X^{n−1}Y^{n−2}.

Note that the polynomial 7.2 is equal to

F′(T)|_{T =2X+Y} = (F_{1}F_{2}F_{3}F_{4})′(T)|_{T =2X+Y} ,

where

F_{1} = T, F_{2} = T − X + Y,

F_{3} = ∏_{i=3}^{n}(T − X + x_{i}) = ∑_{i=0}^{n−2}(T − X)^{n−2−i}(−1)^{i}(X^{i} + … + Y^{i}),

F_{4} = ∏_{i=3}^{n}(T − Y + x_{i}) = ∑_{i=0}^{n−2}(T − Y)^{n−2−i}(−1)^{i}(X^{i} + … + Y^{i}).

We have, further,

F_{3}(T) = (∑_{i=0}^{n−2}(T − X)^{n−2−i}(−1)^{i} \frac{X^{i+1} − Y^{i+1}}{X − Y} ) =

\frac{X(T − X)^{n−2}}{X − Y} (∑_{i=0}^{n−2}(T − X)^{−i}(−1)^{i}X^{i}) −

\frac{Y (T − X)^{n−2}}{X − Y} (∑_{i=0}^{n−2}(T − X)^{−i}(−1)^{i}Y^{i}) =

\frac{X ((T − X)^{n−1} − (−X)^{n−1})}{T (X − Y)} − \frac{Y ((T − X)^{n−1} − (−Y)^{n−1})}{(T − X + Y)(X − Y)}.

We obtain, likewise, that

F_{4}(T) = (∑_{i=0}^{n−2}(T − Y)^{n−2−i}(−1)^{i} \frac{X^{i+1} − Y^{i+1}}{X − Y} ) =

\frac{X(T − Y)^{n−2}}{X − Y} (∑_{i=0}^{n−2}(T − Y)^{−i}(−1)^{i}X^{i}) −

\frac{Y (T − Y)^{n−2}}{X − Y} (∑_{i=0}^{n−2}(T − Y)^{−i}(−1)^{i}Y^{i}) =

\frac{X ((T − Y)^{n−1} − (−X)^{n−1})}{(T + X − Y)(X − Y)} − \frac{Y ((T − Y)^{n−1} − (−Y)^{n−1})}{T (X − Y)}.

We deduce from the latest formula that

F′(T) = \frac{n(X − Y)(T − X)^{n−1} + (−X)^{n} − (−Y)^{n}}{X − Y} ×

\frac{(X − Y)(T − Y)^{n} + T (−X)^{n} − (T + X − Y)(−Y)^{n}}{T (T + X − Y)(X − Y)} +

\frac{(X − Y)(T − X)^{n} + (T − X + Y)(−X)^{n} − T (−Y)^{n}}{T (X − Y)} ×

\frac{(n(T + X − Y) − (T − Y))(T − Y)^{n−1} + (−X)^{n}}{(T + X − Y)^{2}} −

\frac{(X − Y)(T − X)^{n} + (T − X + Y)(−X)^{n} − T (−Y)^{n}}{T (X − Y)} ×

7.2 Degrees of Discriminants of Polarized Flag Varieties 115

\frac{(X − Y)(T − Y)^{n} + T (−X)^{n} − (T + X − Y)(−Y)^{n}}{T (T + X − Y)(X − Y)} .

Using the elementary formulae

\frac{α^{n}(γ − β) + β^{n}(α − γ) + γ^{n}(β − α)}{(α − β)(β − γ)(γ − α)} = ∑_{i+j+k=n−2} α^{i}β^{j}γ^{k},

\frac{α^{n}(γ − β) + β^{n}(α − γ) + γ^{n}(β − α)}{(α − β)(γ − β)} = ∑_{i=0}^{n−2} β^{i}(α^{n−1−i} − γ^{n−1−i}),

\frac{(n(α − β) − α)α^{n−1} + β^{n}}{(α − β)^{2}} = ∑_{i=0}^{n−2}(n − 1 − i)α^{n−2−i}β^{i},

we obtain

F′(T) = (n(T − X)^{n−1} + (−1)^{n} ∑_{i=0}^{n−1} X^{n−1−i}Y^{i})

×( ∑_{i+j+k=n−2}(T − Y)^{i}(−X)^{j}(−Y)^{k})+

(∑_{i=0}^{n−2}(−X)^{i}((T − X)^{n−1−i} − (−Y)^{n−1−i}))

×(∑_{i=0}^{n−2}(n − 1 − i)(T − Y)^{n−2−i}(−X)^{i})−

(∑_{i=0}^{n−2}(−X)^{i}((T − X)^{n−1−i} − (−Y)^{n−1−i}))

×( ∑_{i+j+k=n−2}(T − Y)^{i}(−X)^{j}(−Y)^{k}).

Putting T = aX + bY in the latest formula, we obtain

(∑_{i=0}^{n−1}(1 + (−1)^{n}n(a − 1)^{n−1−i}b^{i}\binom{n − 1}{i})X^{n−1−i}Y^{i})

×(∑_{i=0}^{n−2} [ \begin{matrix}n − 2 \\ i \end{matrix} ]_{a,b−1} X^{i}Y^{n−2−i})−

(∑_{i=0}^{n−1}([ \begin{matrix}n − 1 \\ i \end{matrix} ]_{a−1,b} − [ \begin{matrix}n − 2 \\ i − 1 \end{matrix} ]_{a−1,b} − 1)X^{n−1−i}Y^{i})

×(∑_{i=0}^{n−2}(\{ \begin{matrix}n − 2 \\ i \end{matrix} \}_{a,b−1} − \{ \begin{matrix}n − 3 \\ i \end{matrix} \}_{a,b−1})X^{n−2−i}Y^{i})+

(∑_{i=0}^{n−1}([ \begin{matrix}n − 1 \\ i \end{matrix} ]_{a−1,b} − [ \begin{matrix}n − 2 \\ i − 1 \end{matrix} ]_{a−1,b} − 1)X^{i}Y^{n−1−i})

×(∑_{i=0}^{n−2} [ \begin{matrix}n − 2 \\ i \end{matrix} ]_{a,b−1} X^{i}Y^{n−2−i}),

where

[ \begin{matrix}n \\ k \end{matrix} ]_{xy} = ∑_{p=0}^{k} ∑_{q=0}^{n−k}(−1)^{p+q}x^{p}y^{q}\binom{p + q}{p},

\{ \begin{matrix}n \\ k \end{matrix} \}_{xy} = ∑_{p=0}^{k} ∑_{q=0}^{n−k}(p + q + 1)(−1)^{p+q}x^{p}y^{q}\binom{p + q}{p}.

It remains to calculate the degree of the discriminant, that is, the difference between the coefficients of X^{n−1}Y^{n−2} and X^{n−2}Y^{n−1} in the expression for F′(aX + bY). After some transformations we obtain

∑_{i=0}^{n−1} ((−1)^{n}n(a − 1)^{n−1−i}b^{i}\binom{n − 1}{i} + W_{i}) ×

([ \begin{matrix}n − 2 \\ i \end{matrix} ]_{a,b−1} − [ \begin{matrix}n − 2 \\ i − 1 \end{matrix} ]_{a,b−1}) −

∑_{i=0}^{n−1} (W_{i} − 1) ×

(\{ \begin{matrix}n − 2 \\ i \end{matrix} \}_{a,b−1} − \{ \begin{matrix}n − 3 \\ i \end{matrix} \}_{a,b−1} − \{ \begin{matrix}n − 2 \\ i − 1 \end{matrix} \}_{a,b−1} + \{ \begin{matrix}n − 3 \\ i − 1 \end{matrix} \}_{a,b−1}) ,

where

W_{i} = [ \begin{matrix}n − 1 \\ n − 1 − i \end{matrix} ]_{a−1,b} − [ \begin{matrix}n − 2 \\ n − 1 − i \end{matrix} ]_{a−1,b} .

This is the degree of the discriminant of the irreducible SL_{n}-module with the highest weight (a − b)φ_{1} + bφ_{2}.

Substituting b = 1 in the last formula, we obtain

\frac{(n^{2} − n)a^{n+1} − (n^{2} + n)a^{n−1} − 2n(−1)^{n}}{(a + 1)^{2}} .

⊓ ⊔

7.3 Degree of the Discriminant as a Function in L

Suppose that X is a smooth projective variety. If L is a very ample line bundle on X then we call the pair (X, L) a polarized variety. Any polarized variety admits a canonical embedding X ⊂ ℙ(H^{0}(X, L)^{∗}). Let (X, L)^{∗} ⊂ ℙ(H^{0}(X, L)) be the corresponding dual variety, which parametrises singular divisors in the linear system |L|. If (X, L)^{∗} is a hypersurface then the discriminant Δ_{(X,L)} is a unique (up to a scalar) polynomial defining (X, L)^{∗}. If def(X, L) > 0 then we set Δ_{(X,L)} = 1.

7.3 Degree of the Discriminant as a Function in L 117

7.3.A General Positivity Theorem

We start with the following simple but very useful observation

Theorem 7.15 . Suppose that L, M are very ample line bundles on X. Then def(X, L ⊗ M) = 0.

Proof. Indeed, we have deg Δ_{(X,L⊗M)} = ∫_{X} c_{n}(J(L ⊗ M) by Theorem 7.1. Therefore,

deg Δ_{(X,L⊗M)} = ∫_{X} c_{n}(J(L) ⊗ M) = ∑_{i=0}^{n}(i + 1) ∫_{X} c_{1}(M)^{i} · c_{n−i}(J(L)).

Since J(L) is spanned, all summands are non-negative. Since ∫_{X} c_{1}(M)^{n} > 0 (being equal to the degree of X in the embedding determined by M), the whole sum is positive. ⊓ ⊔

For example, if X is embedded in ℙ(V) and then re-embedded in ℙ(S^{d}V) via the Veronese embedding then the dual variety X^{∗} of this re-embedding is a hypersurface.

Suppose now that L_{1}, …, L_{r} are line bundles on X such that the corresponding linear systems |L_{i}| have no base points and such that for each i there is a representative of the linear system |L_{i}| which is a smooth divisor on X. Suppose further that any line bundle L of the form L = L_{1}^{⊗n_{1}} ⊗ … ⊗ L_{r}^{⊗n_{r}} is very ample for positive n_{i}. Then the degree deg Δ_{(X,L)} is a function in n_{1}, …, n_{r}. Let us introduce new non-negative integers m_{i} = n_{i} − 1. Then deg Δ_{(X,L)} = f(m_{1}, …, m_{r}).

Theorem 7.16 ([CW])

(a) The function f is a non-trivial polynomial with non-negative coefficients.

(b) If each n_{i} ≥ 2 then (X, L)^{∗} is a hypersurface.

Proof. Let n = dim X. Recall that by Kleiman–Katz–Holme formula 7.2 we have

deg Δ_{(X,L)} = ∑_{i=0}^{n}(i + 1) ∫_{X} c_{n−i}(Ω_{X}^{1}) · L^{i} = P′(L),

where

P (t) = ∑_{i=0}^{n} c_{n−i}(Ω_{X}^{1})t^{i+1}.

This clearly implies that f is a polynomial in n_{i} and, hence, in m_{i}. This polynomial is non-trivial by Theorem 7.15. Therefore, we need to check positivity only. Using the Taylor expansion, it suffices to check the following claim. For each j = 1, …, n and for each collection of nonnegative integers u_{1}, …, u_{r} such that u_{1} + … + u_{r} = j − 1, we have

∫_{X} P^{(j)}(L) · L_{1}^{u_{1}} … L_{r}^{u_{r}} ≥ 0.

The conditions imposed on L_{i} imply that any intersection of the form L_{1}^{u_{1}} … L_{r}^{u_{r}} (as an element of a Chow ring) is represented by a smooth subvariety of codimension j − 1. Therefore, it suffices to check that for any such subvariety Y ⊂ X of codimension j − 1 we have

∫_{X} P^{(j)}(L) · [Y] ≥ 0.

Using exact sequence 3.1 it is easy to see that

P^{(j)}(L) = j!c_{n+1−j}(J(L)).

Therefore it remains to show that

∫_{X} c_{n+1−j}(J(L)) · [Y] ≥ 0.

This follows from the fact that J(L) is spanned by its global sections (see [Fu1]). ⊓ ⊔

7.3.B Applications to Polarized Flag Varieties

The conditions of Theorem 7.16 are perfectly well satisfied if X is a flag variety G/P. We shall use notations from 2.1. The fundamental weights ω_{1}, … ω_{k} dual to the simple roots in Π_{G/P} generate the lattice P_{G/P} isomorphic to Pic(G/P). We shall denote the line bundle L_{ω_{i}} by L_{i}. Then any line bundle on G/P have a form L = L_{1}^{⊗n_{1}} ⊗ … ⊗ L_{r}^{⊗n_{r}}. L is very ample if and only if all n_{i} > 0. Then the degree deg Δ_{(X,L)} is a function in n_{1}, …, n_{r}. Let us introduce new non-negative integers m_{i} = n_{i} − 1. Then deg Δ_{(X,L)} = f(m_{1}, …, m_{r}). All conditions of Theorem 7.16 are clearly satisfied, therefore f is a non-trivial polynomial with non-negative coefficients.

Example 7.17 Let V = ℂ^{r}, G = SL(V) and let P be the stabiliser of the line in V. Then G/P = ℙ(V). Take λ = nω_{1}, n > 0. Then L_{1} = O(n). The degree of Δ_{(G/P,λ)} is the degree of the classical discriminant of a homogeneous form of degree n in r variables. Therefore by Example 7.9 we have deg Δ_{(G/P,λ)} = r(n − 1)^{r−1}. Therefore the polynomial f in this case has a form f = rm^{r−1}, so it is a polynomial with nonnegative coefficients.

Example 7.18 Let G = SL_{3} and let P = B be the Borel subgroup. The weight λ can be written as λ = m_{1}ω_{1} + m_{2}ω_{2}. One can check using the Kleiman formula that

deg Δ_{(G/B,λ)} = 12(m_{1}^{2}m_{2} +m_{1}m_{2}^{2})−6(m_{1}^{2} +4m_{1}m_{2} +m_{2}^{2})+12(m_{1} +m_{2})−6.

Substituting n_{1} = m_{1} − 1, n_{2} = m_{2} − 1 we see that

deg Δ_{(G/B,λ)} = 12(n_{1}^{2}n_{2} + n_{1}n_{2}^{2}) + 6(n_{1}^{2} + 4n_{1}n_{2} + n_{2}^{2}) + 12(n_{1} + n_{2}) + 6.

Again, we get a polynomial with nonnegative coefficients.

Example 7.19 Assume that the weight λ satisfies the conditions m_{i} ≥ 2 (is very strictly dominant). Then the results of this section clearly imply that the dual variety (G/P, λ)^{∗} is a hypersurface.

7.4 Gelfand–Kapranov Formula

Theorem 3.9 identifies Δ_{X} with the determinant of any of the discriminant complexes C_{±}^{∗}(X, M), provided the complex is stably twisted. By Corollary 3.6 this implies a formula for deg Δ_{X}.

Theorem 7.20 ([GKZ2]) If the complex C_{−}^{∗}(X, M) is stably twisted then

deg Δ_{X} = ∑_{i=− dim X−1}^{0}(−1)^{i} · i · dim C_{−}^{i}(X, M).

If the complex C_{+}^{∗}(X, M) is stably twisted then

deg Δ_{X} = (−1)^{dim X+1} ∑_{i=0}^{dim X+1}(−1)^{i} · i · dim C_{+}^{i}(X, M).

Thus deg Δ_{X} can be expressed through the dimensions of vector spaces

H^{0} (X, Λ^{−i}J(L)^{∗} ⊗ M) or H^{0} (X, Λ^{i}J(L) ⊗ M) .

It is possible to rewrite these dimensions using simpler quantities associated with X, at least in case M = O_{X}(l), l ≫ 0. For any coherent sheaf F on X, we write

F(l) = F ⊗ O_{X}(l), h^{i}(F) = dim H^{i}(X, F), χ(F) = ∑(−1)^{i}h^{i}(F).

The number χ(F) is called the Euler characteristic of F. It follows from the Riemann-Roch-Hirzebruch theorem [Hir] that χ(F(l)) is a polynomial in l. This polynomial is called the Hilbert polynomial of F and denoted by h_{F}(l). For l ≫ 0, the higher cohomology of F(l) vanish and we have

h_{F}(l) = χ(F(l) = h^{0}(F(l)).

This is a more usual definition of Hilbert polynomials, see [Ha1]. Simple calculations show that Theorem 7.20 can be reformulated as follows. Recall that T_{X} is the tangent bundle of X and Ω_{X}^{i} = Λ^{i}T_{X}^{∗} is the bundle of differential i-forms on X.

Theorem 7.21 ([GKZ2]) For any l ∈ ℤ we have

deg Δ_{X} = ∑_{i=− dim X−1}^{0}(−1)^{i} · i · (h_{Λ^{−i}T_{X}} (i + l) + h_{Λ^{−i−1}T_{X}} (i + l)) ,

deg Δ_{X} = (−1)^{dim X+1} ∑_{i=0}^{dim X+1}(−1)^{i} · i · (h_{Ω_{X}^{i}} (i + l) + h_{Ω_{X}^{i−1}}(i + l)) .

This theorem was generalized in [GK] to handle the case of the positive def X. For any l ∈ ℤ consider the polynomial f_{l}(q) in a formal variable q

f_{l}(q) = ∑_{i=0}^{dim X+1}(−1)^{i} (h_{Λ^{i}T_{X}} (i + l) + h_{Λ^{i−1}T_{X}} (i + l)) q^{i},

where all polynomials that have no sense are supposed to be equal to 0. Clearly f_{l}(q) is also a polynomial in l.

Theorem 7.22 ([GK, GKZ2]) Let l ∈ ℤ_{+} be any non-negative integer. The codimension of X^{∗} equals the order of the zero of f_{l}(q) at q = 1. Let this order be μ and let f_{l}(q) = a_{μ}(l)(1 − q)^{μ} + O((1 − q)^{μ+1}). Then a_{μ}(l) = \binom{μ+l−1}{μ−1} · deg X^{∗}.

8. Milnor Classes and Multiplicities of Discriminants

Preliminaries

The notion of the Milnor class was introduced in [Al] generalizing Milnor numbers of isolated singularities [Mi] and Parusiǹski μ-numbers [Pa1]. We review some applications of this technique to dual varieties, in particular for the description of their singularities. We also give other results in this direction. In this chapter we assume that the reader is more familiar with modern intersection theory than is necessary for other chapters of this book. But to illustrate how the topology can be used for study of dual varieties, we give an elementary proof of the class formula in the first section (cf. Example 7.5).

8.1 Class Formula

Suppose that X ⊂ ℙ^{N} is a smooth projective n-dimensional variety with the dual variety X^{∗} ⊂ (ℙ^{N})^{∗}. Consider a generic line L ⊂ (ℙ^{N})^{∗} such that the intersection L ∩ X^{∗} consists of d smooth on X^{∗} points if def X = 0, where d is the degree of X^{∗}. If def X > 0 then this intersection is empty and we set d = 0.

L can be considered as a pencil (one-dimensional linear system) of divisors on X, therefore it defines a rational map F : X → ℙ^{1}. More precisely, if L is spanned by hyperplanes H_{1} and H_{2} with equations f_{1} = 0 and f_{2} = 0 then F has a form x ↦ (f_{1}(x) : f_{2}(x)) for any x ∈ X. Thus F is not defined along the subvariety X ∩H_{1} ∩H_{2} of codimension 2. We shall blow it up and get the variety X̃ and the regular morphism F̃ : X̃ → ℙ^{1}. Let D ⊂ X̃ be a preimage of X ∩ H_{1} ∩ H_{2}.

Now let us calculate the topological Euler characteristic χ(X̃) in two ways. First,

\begin{matrix}χ(X̃) = χ(X̃ \ D) + χ(D) = χ(X \ X ∩ H_{1} ∩ H_{2}) + χ(D) = \\ χ(X \ X ∩ H_{1} ∩ H_{2}) + 2χ(X ∩ H_{1} ∩ H_{2}) = χ(X) + χ(X ∩ H_{1} ∩ H_{2}). \end{matrix}

Here we use the fact that D is a ℙ^{1}-bundle over X ∩ H_{1} ∩ H_{2}, therefore χ(D) = 2χ(X∩H_{1}∩H_{2}). Notice that the blowing up is an isomorphism on the complement of the exceptional divisor, hence χ(X̃ \D) = χ(X \X ∩H_{1} ∩H_{2}).

On the other hand, we may use F̃ to calculate χ(X̃). For x ∈ ℙ^{1}, we have χ(F̃^{−1}(x)) = χ(X ∩ H_{x}), where H_{x} is a hyperplane corresponding to x. Let x_{1}, …, x_{d} ∈ ℙ^{1} be points that correspond to the intersection of L with X^{∗}. Then for any x ∈ ℙ^{1} \ {x_{1}, …, x_{d}}, X ∩ H_{x} is a smooth divisor. For any x ∈ {x_{1}, …, x_{d}}, X ∩ H_{x} has a simple quadratic singularity (see Theorem 6.12). It is clear then that χ(X ∩ H_{x}) = χ(X ∩ H_{y}) = χ(X ∩ H), where x, y ∈ ℙ^{1} \ {x_{1}, …, x_{d}} and H is a generic hyperplane. For x ∈ {x_{1}, …, x_{d}}, near the simple quadratic singularity of X ∩ H_{x}, the family of divisors X ∩ H_{y}, y → x, looks like the family of smooth (n − 1)-dimensional quadrics Q_{ε} with equations T_{1}^{2} +…+T_{n−1}^{2} = εT_{n}^{2} near the unique singular point (0 : …: 0 : 1) of the quadric Q_{0} with the equation T_{1}^{2} + … + T_{n−1}^{2} = 0. Therefore,

χ(X̃) = χ(ℙ^{1})χ(X ∩H)+d(χ(Q_{0})−χ(Q_{1})) = 2χ(X ∩H)+d(χ(Q_{0})−χ(Q_{1})).

Combining two formulas for χ(X̃), we get

χ(X) + χ(X ∩ H_{1} ∩ H_{2}) = 2χ(X ∩ H) + d(χ(Q_{0}) − χ(Q_{1})).

The Euler charachteristic of k-dimensional smooth quadric is equal to k + 2 for k even and to k+1 for k odd. The Euler charachteristic of a k-dimensional quadric with a unique singularity is equal to k + 1 for k even and to k + 2 for k odd, since it is a cone over (k − 1)-dimensional quadric. Finally, we obtain the class formula

d = (−1)^{n} [χ(X) − 2χ(X ∩ H) + χ(X ∩ H ∩ H′)] .

This formula has a large number of applications, see Example 7.5, 7.6, 7.7, 7.8, etc. In this chapter we consider various attempts to generalise this formula in different directions.

8.2 Milnor Classes

Suppose that M is a smooth n-dimensional algebraic variety, L is a line bundle on M, and X is the zero-scheme of a section of L. The singular scheme of X, Sing X, is the scheme supported on the singular locus of X, and defined locally by the Jacobian ideal (\frac{∂F}{x_{1}} , …, \frac{∂F}{x_{n}} ), where x_{1}, …, x_{n} are local parameters for M, and F is the section of L defining X.

Definition 8.1 Let Y be the singular scheme of a section of a line bundle L on a smooth variety M. The Milnor class of Y with respect to L is the class

μ_{L}(Y) = c(T^{∗}M ⊗ L) ∩ s(Y, M)

in the Chow group A_{∗}Y of Y.

Here c denotes the total Chern class and s is the Segre class (in the sense of [Fu1]). The following theorem summarizes the main properties of the Milnor class. The reader may consult [Al] for proofs and further details.

Theorem 8.2 ([Al])

(a) The Milnor class μ_{L}(Y) only depends on Y and L|_{Y}. That is, the definition of the Milnor class is independent on the choice of a smooth ambient variety M in which Y is realized as the singular scheme of a section of a line bundle restricting to L|_{Y}.

(b) Suppose that L is generated by global sections and let X_{g} be a generic section of L. Then

μ_{L}(Y ∩ X_{g}) = c_{1}(L) ∩ μ_{L}(Y).

(c) Suppose that Y is non-singular. Then

μ_{L}(Y) = c(T^{∗}Y ⊗ L) ∩ [Y].

Example 8.3 If Y is supported on a point P, then μ_{L}(Y) = m[P], where m is the Milnor number of X at P. Indeed, in this case μ_{L}(Y) = s(Y, M), so m is the coefficient of [P] in s(Y, M), which agrees with the definition of the Milnor number (see [Fu1]).

Example 8.4 Even if dim Y > 0, let |Y | be the support variety of Y, assume that Y is complete. Then in [Pa1] there is introduced an invariant, the μ-number of X at |Y |, agreeing with the ordinary Milnor number in case |Y | is a point. We claim that in fact this number equals the degree of the 0-dimensional component of μ_{L}(Y). Namely, the differential of the local equations of X in M determines a section of (T^{∗}M ⊗ L)|_{X}. This can be extended to a holomorphic section s_{X} of T^{∗}M ⊗ L over the whole of M. Then the μ-number is defined in [Pa1] as the contribution of |Y | to the intersection of s_{X} with the zero section of T^{∗}M ⊗ L. In the neighborhood of |Y | we have the fiber diagram

Y ⟶  M
↓  ↓^{s_{X}}
M ⟶^{s_{0}} T^{∗}M ⊗ L

that is, Y is the scheme-theoretic intersection of the two sections, so the contribution of |Y | to the intersection number of the sections is equal (see [Fu1]) the degree of the 0-dimensional component of

c(T^{∗}M ⊗ L) ∩ s(Y, M),

which is the claim. In fact, in view of Theorem 8.3, the numerical information carried by the Milnor class of Y (when Y is complete) is essentially equivalent to μ-numbers of |Y | and its generic sections |Y ∩ X_{g}|, |X ∩ X_{g_{1}} ∩ X_{g_{2}}|, . . ..

8.3 Applications to Multiplicities of Discriminants

We shall now apply the machinery of the previous section to our usual situation. Suppose that X ⊂ ℙ^{N} is a smooth projective variety, L = O_{X}(1), X^{∗} ⊂ ℙ^{N}^{∗} is the dual variety parametrizing hyperplanes H such that the singular scheme (or the contact scheme in this case) Sing X ∩ H is not empty. Then according to the previous section we have the Milnor class μ_{L}(Sing X ∩ H) ∈ A_{∗}(Sing X ∩ H). The problem that we address here is how to calculate the multiplicity of X^{∗} at H in terms of the Milnor class.

Theorem 8.5 ([Al, AC, Di1, N, Pa2]) Let X ∩ H be any singular hyperplane section of X. Then

(a) The defect of the dual variety X^{∗} is the smallest integer r ≥ 0 such that

∫ c_{1}(L)^{r}c(L) ∩ μ_{L}(Sing X ∩ H) ≠ 0;

and for r = def X this number equals the multiplicity m_{H}X^{∗} of X^{∗} at H.

(b) If def X = r then m_{H}X^{∗} is equal to

∫ (−1)^{r}μ_{L}(Sing X ∩ H) + c_{1}(L)^{r+1} ∩ μ_{L}(Sing X ∩ H).

Proof. (a) Induction on r. If r = 0 then the integral is equal to

∫ c(L) ∩ μ_{L}(Sing X ∩ H) = ∫ c(L)c(T^{∗}X ⊗ L) ∩ s(Sing X ∩ H, X)

= ∫ c((T^{∗}X ⊕ O) ⊗ L) ∩ s(Sing X ∩ H, X).

Thinking of H as a point of X^{∗}, we have s(H, X^{∗}) = (m_{H}X^{∗})X^{∗}, thus

m_{H}X^{∗} = deg s(H, X^{∗}).

The dual variety is the projection from the conormal variety I_{X}, X^{∗} = π(I_{X}). Since r = 0, the degree of this projection is equal to 1. The fiber Y of π over H is identified with the contact locus Sing H ∩ X (as a scheme). Therefore, we have s(H, X^{∗}) = π_{∗}(Y, I_{X}). Taking degrees, we get

m_{H}X^{∗} = deg s(Y, I_{X}).

Now using the second projection I_{X} → X, which is the projective bundle, it is quite easy to see that

s(Y, I_{X}) = c(J(L)) ∩ s(Sing X ∩ H, X),

where J(L) is the jet bundle of L. It remains to notice that c(J(L)) = c((T^{∗}X ⊕ O) ⊗ L) by the exact sequence (3.1).

Now let r > 0. By the similar argument as above, we have ∫ c(L) ∩ μ_{L}(Sing X ∩ H) = 0. By Theorem 1.30, if X^{∗} is not a hypersurface then the dual variety X′^{∗} of a generic hyperplane section X′ = X ∩ H_{g} is the cone over X^{∗} with H_{g} as a vertex. The multiplicity of X^{∗} at H then evidently equals the multiplicity of X′^{∗} at H; but def X′ = def X − 1, so the statement follows by induction because

∫ c_{1}(L)^{r}c(L) ∩ μ_{L}(Sing X ∩ H)
= ∫ c_{1}(L)^{r−1}c(L) ∩ (c_{1}(L) ∩ μ_{L}(Sing X ∩ H))
= ∫ c_{1}(L)^{r−1}c(L) ∩ μ_{L}(Sing X′ ∩ H),

where we used Theorem 8.2 in the last equality.

(b) By (a) we see that def X ≥ r if and only if the components of dimension i, 0 ≤ i < r of c(L) ∩ μ_{L}(Sing X ∩ H) vanish, for all hyperplane sections X ∩ H. That is, if def X ≥ r, then for all hyperplane sections X ∩ H

c(L) ∩ μ_{L}(Sing X ∩ H) = A_{dim Sing X∩H} + … + A_{r}

with A_{j} a class in dimension j, j = r, …, dim Sing X ∩ H (depending on H). Therefore,

μ_{L}(Sing X ∩ H) = c(L)^{−1} ∩ (A_{dim Sing X∩H} + … + A_{r})

(with A_{r} ≠ 0), and

∫ (−1)^{r}μ_{L}(Sing X ∩ H) = ∑_{i=r}^{dim Sing X∩H}(−1)^{i−r}c_{1}(L)^{i} ∩ A_{i}

while

∫ c_{1}(L)^{r+1}μ_{L}(Sing X ∩ H) = ∑_{i=r+1}^{dim Sing X∩H}(−1)^{i−r+1}c_{1}(L)^{i} ∩ A_{i}.

Hence,

∫ (−1)^{r}μ_{L}(Sing X ∩ H) + c(L)^{r+1} ∩ μ_{L}(Sing X ∩ H) = ∫ c_{1}(L)^{r} ∩ A_{r},

which coincides with the formula in (a). ⊓ ⊔

In case def X = 0 and Sing X ∩ H is finite, the theorem takes the most simple form:

Corollary 8.6 ([Di1, N]) If def X = 0, H ∈ X^{∗}, and Sing X ∩ H is finite we have

m_{H}X^{∗} = ∑_{p∈Sing X∩H} μ(X ∩ H, p),

the sum over ordinary Milnor numbers. In particular, if Sing X ∩ H is finite then H is smooth on X^{∗} if and only if Sing X ∩ H = {p_{0}} and the Hessian of the function f defining X ∩ H is non-zero at p_{0}.

Remark 8.7 An easy Chern class computation shows that ranks (see Section 7.1) could be expressed via Milnor classes as follows:

δ_{i} = ∫_{X} c_{1}(L)^{i}c(L) ∩ μ_{L}(X).

Moreover, Theorem 7.11 can be viewed as a particular case of Theorem 8.5 applied to the zero-section of H^{0}(X, L).

8.4 Multiplicities of the Dual Variety of a Surface

Let X be a smooth projective algebraic surface, L a very ample line bundle on X, X^{∗} ⊂ |L| the dual variety, consisting of singular members of |L|. By Example 6.16, X^{∗} is a hypersurface. We want to determine, for each D ∈ |L|, the multiplicity m_{D}X^{∗} of the hypersurface X^{∗} at the point D. Suppose that the divisor

D = ∑_{i=1}^{r} n_{i}D_{i},

where any divisor D_{i} is reduced and irreducible. Then we have D_{red} = ∑ D_{i}.

Theorem 8.8 ([Al]) With notation as above, we have

m_{D}X^{∗} = (D − D_{red}) · (K_{X} + 2D + D_{red}) + ∑_{p∈D_{red}} μ(D_{red}, p),

where K_{X} is the canonical divisor of X and μ stands for the ordinary Milnor number.

Proof. Take a general H ∈ |L| (i.e. a H intersecting D_{red} transversally) and denote by L ⊂ |L| the pencil containing D and H. Then

m_{D}X^{∗} = deg X^{∗} − s,  (8.1)

where s is the number of singular members of L different from D (each of this singular members has one node as a singular locus). In order to determine s we shall blow up X to construct a family parametrized by L and use Lefschetz’s formula [GH1].

For each i (1 ≤ i ≤ r) denote by p_{ij} (1 ≤ j ≤ H · D_{i}) the points of intersection of H and D_{i}. Let X̂ denote the surface obtained from X by blowing up each point p_{ij} n_{i} times (in the direction of H). The induced pencil L̂ on X̂ is base point free and gives a map f : X̂ → ℙ^{1}. If E_{ij}^{k}, k = 1, …, n_{i} are the exceptional divisors at p_{ij} then the fiber of f at the point 0 (corresponding to D) is

f^{∗}(0) = D + ∑_{i,j} ∑_{1≤k≤n_{i}−1}(n_{i} − k)E_{ij}^{k}.

In other words, the special fiber is isomorphic to D with strings of ℙ^{1}’s (each ℙ^{1} with a certain multiplicity) attached at the points p_{ij}; each string has n_{i} − 1 components.

We now denote

D′ = (f^{∗}(0))_{red} = D_{red} + ∑_{i,j} T_{ij}

the reduced fiber of f at 0, where T_{ij} = ∑_{1≤k≤n_{i}−1} E_{ij}^{k} is the reduced string attached at p_{ij}. Then the standard argument involving Lefschetz’s formula (see [GH1]) gives

χ(X) = 2χ(H) − H · H + (χ(D′) − χ(H)) + ∑_{1≤λ≤s}(χ(H_{λ}) − χ(H)),

where χ denotes the topological Euler characteristic and H_{λ} are the singular fibers of f for λ ≠ 0. Since χ(H_{λ})−χ(H) = 1 (see (8.4) below) and deg X^{∗} = χ(X) − 2χ(H) + H · H (Example 7.8), we can rewrite (8.1) as

m_{D}X^{∗} = χ(D′) − χ(H).  (8.2)

In order to compute χ(D′), denoting T = ∩_{i,j}T_{ij} we have

χ(D′) = χ(D_{red} ∩ T) = χ(D_{red}) + χ(T) − χ(D_{red} ∩ T)  (8.3)

= χ(D_{red}) + ∑_{i,j} χ(T_{ij}) − ∑_{i,j} χ({p_{ij}})

= χ(D_{red}) + ∑_{i,j} n_{i} − ∑_{i,j} 1 = χ(D_{red}) + D · (D − D_{red}).

Now we compute χ(D_{red}). Let Z = ∑_{1≤i≤r} Z_{i} be a reduced connected curve with normalization

ρ : Z̃ = ⊔_{1≤i≤r} Z̃_{i} → Z.

If p ∈ Z is a singular point, denote B(p) = ρ^{−1}(p) the set of branches at p. Topologically, Z is obtained from the smooth real surface Z̃ by identifying each of the sets B(p) to a point p. Therefore, we have

χ(Z̃) − χ(Z) = ∑_{p∈Z}(b(Z, p) − 1),  (8.4)

where b(Z, p) is the number of branches of Z at p. Also, from the exact sequence of sheaves

0 → O_{Z} → ρ_{∗}OZ̃ → ρ_{∗}OZ̃/O_{Z} → 0

we obtain

1 − r + ∑_{p∈Z} δ(Z, p) − h^{1}(O_{Z}) + h^{1}(OZ̃) = 0,

where we set δ(Z, p) = length_{p}(ρ_{∗}OZ̃/O_{Z}), Combining this with (8.4) we obtain

χ(Z) = 2 − 2p_{a}(Z) + ∑_{p∈Z} μ(Z, p),  (8.5)

where

μ(Z, p) = 2δ(Z, p) − b(Z, p) + 1

is the Milnor number of (Z, p). Now writing together (8.2), (8.3), and (8.5) we get

m_{D}X^{∗} = χ(D_{red}) − χ(H) + D × (D − D_{red})

= (2 − 2p_{a}(D_{red})) − (2 − 2p_{a}(H) + D · (D − D_{red}) + ∑_{p∈D_{red}} μ(D_{red}, p)

= −(K_{X} + D_{red}) · D_{red} + (K_{X} + H) · H

+D · (D − D_{red}) + ∑_{p∈D_{red}} μ(D_{red}, p).

After rearranging we finally obtain the claim of the Theorem. ⊓ ⊔

Example 8.9 Suppose that X = ℙ^{2}, L = O_{X}(d). Then X^{∗} parametrizes plane singular curves of degree d. If C is such curve then applying Theorem 8.8 we get

m_{C}X^{∗} = (3(d − 1) − d′) d′ + μ,

where d′ is the degree of C − C_{red} and μ is the sum of the Milnor numbers of the singularities of C_{red}. For example, the multiplicity of the discriminant of plane conics at a double line is 2.

8.5 Further Applications to Dual Varieties

Theorem 6.15 shows that if def X > 0 then the defect and the dimension have the same parity. Since the defect equals the dimension of a contact locus with a generic tangent hyperplane, this can be reformulated by saying that if the generic contact locus is positive-dimensional then it has the same parity as dim X. It is possible to generalize this result for not necessarily generic contact loci using the machinery of Milnor classes. The proof of the following theorem can be found in [Al].

Theorem 8.10 ([Al]) Let X ⊂ ℙ^{N} be a smooth projective variety. Let H ∈ X^{∗} be such that the contact locus Y = Sing X ∩ H is pure-dimensional and non-singular in a neighborhood of some complete curve C ⊂ Y of odd degree. Then

dim Y ≡ dim X mod 2.

For example, if Y is a generic contact locus then Y = ℙ^{def X}. Therefore, if Y is not a point then it contains a line C and we get Theorem 6.15.

The following result follows quite immediately from Theorem 8.5:

Theorem 8.11 ([Al]) Let X ⊂ ℙ^{N} be a smooth projective variety. Suppose that the contact scheme Sing X ∩ H of a hyperplane H with X is a projective space ℙ^{r}. Then the defect def X is equal to r and the dual variety X^{∗} is smooth at H.

Recall that by the contact scheme we mean the contact locus endowed of the scheme structure defined by the Jacobian ideal. The surprising element of this result is that we are not assuming the hyperplane to be generic. The dimension of the contact scheme of the hyperplane need not be equal the dimension of the contact locus of the general hyperplane. The claim, however, is that it is so if the contact scheme is a projective subspace as a scheme. Moreover, it can be shown (also using Theorem 8.5) that if the contact scheme of a variety with a given hyperplane is smooth then the defect of the variety equals the defect of the contact scheme. This ‘explains’ the previous theorem, because the defect of a projective subspace equals its dimension.

In case def X = 0 Theorem 8.11 can be formulated a little bit more explicitly. Let X ⊂ ℙ^{n} be a smooth variety with dual variety X^{∗} being a hypersurface. Let I_{X} ⊂ ℙ^{n} × ℙ^{n}^{∗} be an conormal variety. We denote by π the projection I_{X} → X^{∗}. Let U ⊂ X^{∗} be the set of all points, where the projection π is unramified. Clearly, U is non-empty and open in X^{∗}. In other words, H ⊂ U if the contact locus Sing X ∩ H consists of a single point p scheme-theoretically (the Hessian of the function f defining X ∩H is non-zero at p). Then we have the following

Theorem 8.12 ([Ka]) Suppose that X ⊂ ℙ^{N} is smooth and X^{∗} is a hypersurface. U consists of smooth points of X^{∗} and is the biggest open set in X^{∗} for which the projection π : π^{−1}(U) → U is an isomorphism.

In fact, one may expect that U = X_{sm}. For example, this is so if X^{∗} \ U has codimension 1.

Proposition 8.13 Suppose that X ⊂ ℙ^{N} is smooth and X^{∗} is a hypersurface. If Z = X^{∗} \ U is a hypersurface in X^{∗} (possibly reducible) then Z = X^{∗}_{sing}.

Proof. By a previous theorem X^{∗}_{sing} ⊂ Z. We need to prove an opposite inclusion. Let Y ⊂ X^{∗} be the variety of all points x ∈ X^{∗} such that dim π^{−1}(x) > 0. Then Y ∩ U = ∅, therefore codim_{X^{∗}} Y ≥ 2. Therefore, if Z_{0} ⊂ Z is an irreducible component then for generic point H ∈ Z_{0} the contact locus Sing H ∩X consists of finitely many points. Therefore, H ∈ X^{∗}_{sing} by Corollary 8.6. ⊓ ⊔

The description of the singular locus of the dual variety X^{∗} is known only in some special cases. The most detailed study was performed for hyperdeterminants in [WZ]. To formulate this result, we’ll need the following notations. Suppose first that X ⊂ ℙ^{N} is a smooth variety. We shall define two subvarieties X^{∗}_{cusp} and X^{∗}_{node} in X^{∗}. The variety X^{∗}_{node} is the closure of the set of hyperplanes H such that H is tangent to X at two distinct points. In other words, X^{∗}_{node} = \overline{pr_{1}(I_{X}^{2})}, where

I_{X}^{2} = {(H, x, y) ∈ X^{∗} × X × X | x ≠ y, T̂_{x}X, T̂_{y}X ∈ H}.

The variety X^{∗}_{cusp} consists of all hyperplanes H such that there exists a point x ∈ Sing X ∩ H that is not a simple quadratic singularity. That is, the Hessian of the function f defining X ∩ H is equal to zero at x. Suppose now that X = ℙ^{k_{1}} × … ℙ^{k_{r}} in the Segre embedding. By Theorem 6.8 X^{∗} is a hypersurface if and only if

k_{j} ≤ ∑_{i≠j} k_{i}, j = 1, …, r.

In this case the defining equation of X^{∗} is called the hyperdeterminant of the matrix format (k_{1} + 1) × … × (k_{r} + 1). Without loss of generality we may suppose that k_{1} ≥ k_{2} ≥ … ≥ k_{r}. Then in fact we have only one inequality, namely k_{1} ≤ k_{2} + … + k_{r}. We call the format (k_{1} + 1) × … × (k_{r} + 1) boundary if k_{1} = k_{2} + … + k_{r} and interior if k_{1} < k_{2} + … + k_{r}. For the hyperdeterminants, the variety X^{∗}_{node} can be decomposed further. Namely, for any subset J ⊂ \{1, …, r\} we set X^{∗}_{node}(J) = \overline{pr_{1}(I_{X}^{2}(J))}, where

I_{X}^{2}(J) = {(H, x, y) ∈ I_{X}^{2} | x^{(j)} = y^{(j)} if and only if j ∈ J}.

The proof of the following theorem can be found in [WZ].

Theorem 8.14 ([WZ]) Suppose that k_{1} ≤ k_{2} + … + k_{r}, k_{1} ≥ … ≥ k_{r}.

(a) If the format is boundary then X^{∗}_{sing} is an irreducible hypersurface in X^{∗}. Furthermore, we have X^{∗}_{sing} = X^{∗}_{node}(∅) if the format is different from 4 × 2 × 2. In this exceptional case X^{∗}_{sing} = X^{∗}_{node}(1).

(b) If the format is interior and does not belong to the following list of exceptions, then X^{∗}_{sing} has two irreducible components X^{∗}_{cusp} and X^{∗}_{node}(∅), both having codimension 1 in X^{∗}. The list of exceptional cases consists of the following 3- and 4-dimensional formats:

(1) For 2 × 2 × 2 matrices X^{∗}_{sing} = X^{∗}_{cusp} and has three irreducible components, all of codimension 2 in X^{∗}.

(2) For 3 × 2 × 2 matrices X^{∗}_{sing} = X^{∗}_{cusp} and is irreducible.

(3) For 3×3×3 matrices X^{∗}_{sing} has five irreducible components X^{∗}_{cusp}, X^{∗}_{node}(∅), X^{∗}_{node}(1), X^{∗}_{node}(2), X^{∗}_{node}(3).

(4) For m × m × 3 matrices with m > 3 X^{∗}_{sing} has three irreducible components X^{∗}_{cusp}, X^{∗}_{node}(∅), and X^{∗}_{node}(3).

(5) For 2 × 2 × 2 × 2 matrices X^{∗}_{sing} has eight irreducible components X^{∗}_{cusp}, X^{∗}_{node}(∅), and X^{∗}_{node}(i, j) for 1 ≤ i < j ≤ 4.

(6) For m × m × 2 × 2 matrices with m > 2 X^{∗}_{sing} has three irreducible components X^{∗}_{cusp}, X^{∗}_{node}(∅), and X^{∗}_{node}(3, 4).

In each of the cases (2) to (6) all irreducible components of X^{∗}_{sing} have codimension 1.

132 8. Milnor Classes and Multiplicities of Discriminants

9. Mori Theory and Dual Varieties

Preliminaries

One of the basic ideas of the minimal model program (see e.g. [CKM, KMM, Ko, Wi]) is to study projective varieties and their morphisms by understanding the behaviour of rational curves on these varieties. Since projective varieties with positive defect are covered by projective lines, it is not very surprising that the machinery of the minimal model program can be used to study them. This approach was implemented in a remarkable serie of papers by several authors, starting from [BSW]. In this chapter we give a survey of these results. In section 9.1 we give all necessary definitions and facts from the minimal model program. In section 9.2 we explain how to apply this machinery to study projective varieties with positive defect. Most results of this section are due to [BSW, BFS1, Sn]. We also give a list of smooth projective varieties X with positive defect such that dim X ≤ 10. The study of these varieties was initiated in [E1, E2], continued in [LS] and finished in [BFS1], with many contributions of other people. Though the natural framework for this theory is the class of smooth projective varieties, recently [LPS] many of these results where extended to projective varieties with mild singularities, say with smooth normalization.

Polarized flag varieties with positive defect were first classified in [KM] using Theorem 6.1. Later a much more quicker approach was suggested in [Sn]. The idea is to use the machinery of the minimal model program. In Section 9.3 we give the classification of polarized flag varieties with positive defect.

9.1 Some Results From Mori Theory

Divisors. Let X be a smooth projective variety. If H is a very ample Cartier divisor on X (hyperplane section of some projective embedding) then a pair (X, H) will be called a polarized variety. A divisor D on X is called semiample if the linear system |mD| is base-point-free for some m ≫ 0. Let K_{X} denote a canonical divisor.

We denote by Div(X) a free abelian group of Cartier divisors on X. If two divisors are linearly equivalent (differ by a principal divisor) it will be denoted by D_{1} ∼ D_{2}. Pic(X) is the group of algebraic linear bundles on X. It is naturally isomorphic to a factorgroup of Div(X) modulo principal divisors. For any D ∈ Div(X) the corresponding line bundle will be denoted by O(D).

1-cycles. A 1-cycle on X is a finite sum

Z = ∑ n_{i}C_{i}

of irreducible curves C_{i} ⊂ X with integer multiplicities n_{i}. A 1-cycle Z is called effective if all n_{i} ≥ 0.

The intersection product. If D is a Cartier divisor on X and C ⊂ X is an irreducible curve then we define an intersection product D · C ∈ ℤ as deg g^{∗}O(D)|_{C}, where g : C̄ → C is a normalisation. This product can be extended by linearity to a ℤ-valued pairing of free abelian groups Div(X) and a group of 1-cycles. This pairing is degenerate. A 1-cycle C (resp. a Cartier divisor D) is called numerically trivial if D · C = 0 for any Cartier divisor D (resp. for any 1-cycle C). Two 1-cycles are numerically equivalent, Z_{1} ≡ Z_{2}, if Z_{1} − Z_{2} is numerically trivial. Dually, two Cartier divisors are numerically equivalent, D_{1} ≡ D_{2}, if D_{1} − D_{2} is numerically trivial. The group N_{ℤ}^{1}(X) of Cartier divisors modulo numerical equivalence is a factorgroup of the Neron– Severi group NS(X) of Cartier divisors modulo algebraic equivalence, which is in turn isomorphic to a subgroup of H^{2}(X, ℤ) by a standard argument involving the exponential exact sequence of sheaves

0 → ℤ → O_{X} → O_{X}^{∗} → 0.

Since H^{2}(X, ℤ) is a finitely generated abelian group it follows that N_{ℤ}^{1}(X) is a free abelian group of finite rank ρ(X), this rank is called a Picard number of X. Therefore, the group N_{1}^{ℤ}(X) of 1-cycles modulo numerical equivalence is also a free abelian group of rank ρ(X).

The cone of effective 1-cycles. We consider two real vector spaces of dimension ρ(X):

N_{1}(X) = N_{1}^{ℤ}(X) ⊗ ℝ, N^{1}(X) = N_{ℤ}^{1}(X) ⊗ ℝ.

These vector spaces are naturally dual to each other by means of the intersection product. The main combinatorial object of the minimal model program is defined as follows:

Definition 9.1

– The cone of effective 1-cycles NE_{1}(X) ⊂ N_{1}(X) is a cone generated by classes of effective 1-cycles.

– \overline{NE}_{1}(X) is its closure in N_{1}(X).

– The positive part NE_{+}(X) of \overline{NE}_{1}(X) is defined as

NE_{+}(X) = \{Z ∈ \overline{NE}_{1}(X) | K_{X} · Z ≥ 0\}.

Nef divisors. For D ∈ N^{1}(X), we say that D is nef (or numerically effective) if D · Z ≥ 0 for any Z ∈ NE_{1}(X) (or equivalently for any Z ∈ \overline{NE}_{1}(X)). Therefore, nef vectors form a closed cone NE^{1}(X) in N^{1}(X), dual to the cone \overline{NE}_{1}(X). For example, if D is an ample divisor then for any curve C ⊂ X the intersection number C · D > 0. Therefore, D is nef. In fact, we have the following Kleiman criterion:

Theorem 9.2 ([Kl6]) D is ample iff D · Z > 0 for any Z ∈ \overline{NE}_{1}(X), Z ≠ 0.

Therefore, ample divisors correspond to the interior of the cone NE^{1}(X). Clearly, all introduced cones are non-degenerate (are not contained in any

hyperplane).

ℚ-divisors. We define ℚ-Cartier divisors as elements of a ℚ-vector space Div(X) ⊗ ℚ. The ℚ-Cartier divisor is called nef if the corresponding element in N^{1}(X) is nef. The ℚ-Cartier divisor is called ample if some multiple of it is ample in a usual sense. In the sequel we shall call ℚ-Cartier divisors simply divisors. This will not lead to any confusion.

Extremal rays. An extremal ray is a half line R in \overline{NE}_{1}(X) such that

– K_{X} · R < 0 (K_{X} · Z < 0 for any non-zero Z ∈ R);

– R is a face of \overline{NE}_{1}(X), i.e. for any Z_{1}, Z_{2} ∈ \overline{NE}_{1}(X) if Z_{1} + Z_{2} ∈ R then Z_{1} and Z_{2} are in R.

An extremal rational curve C is a rational curve on X such that ℝ_{+}[C] is an extremal ray and −K_{X} · C ≤ dim X + 1. The following is a fundamental result of Mori Theory, usually referred to as the Mori Cone Theorem:

Theorem 9.3 ([Mor]) The cone \overline{NE}_{1}(X) is the smallest convex cone containing NE_{+}(X) and all the extremal rays. For any open convex cone U containing NE_{+}(X) \ {0} there are finitely many extremal rays that do not lie in U ∪ {0}. Every extremal ray is spanned by an extremal rational curve.

Another fundamental result in Mori theory is the Kawamata–Shokurov Contraction Theorem:

Theorem 9.4 ([Kaw]) Let D be a numerically effective divisor on X. If, for some a ≥ 1, the divisor aD − K_{X} is nef and big (i.e. (aD − K_{X})^{n} > 0), then D is semi-ample.

One of the most useful forms of the Contraction Theorem is the following

Theorem 9.5 Let R be an extremal ray in \overline{NE}_{1}(X). Then there exists a normal projective variety Y and a surjective morphism with connected fibers contr_{R} : X → Y with connected fibers such that for any curve C on X, contr_{R} contracts C to a point if and only if [C] ∈ R.

Proof. Indeed, let D ∈ NE^{1}(X) be such that D · R = 0 and D · Z > 0 for any Z ∈ \overline{NE}_{1}(X) \ R. It follows from the Mori Cone Theorem that we may assume D is a divisor. (In fact, usually this theorem is proved as a part of the proof of the Mori Cone Theorem. Then the fact that D can be chosen to be a divisor is deduced from the Rationality Theorem, see below.) The Kleiman Criterion applies that for a ≫ 0 the divisor aD − K_{X} is ample. Therefore, D is semi-ample by the Contraction Theorem. ⊓ ⊔

The nef value and the nef morphism. Suppose that (X, H) is a smooth polarized projective variety. Assume that K_{X} is not nef.

Definition 9.6

τ = min{t ∈ ℝ | K_{X} + tH is nef}

is called a nef value of (X, H).

If X is fixed then τ will be called a nef value of H, or a nef value of L, where L is a line bundle corresponding to H. Clearly 0 < t < ∞. Notice also that τ is a nef value of (X, H) if and only if K_{X} + τH is nef, but not ample. The following theorem is known as the Kawamata Rationality Theorem:

Theorem 9.7 ([Kaw]) The nef value τ is a rational number.

The next theorem is a particular case of the Kawamata–Shokurov Contraction Theorem

Theorem 9.8 Suppose that τ is a nef value of (X, H). The linear system |m(K_{X} + τH)| is base point free for m ≫ 0. It defines a regular morphism Φ : X → Y onto a normal variety Y with connected fibers. Φ is called a nef value morphism.

Length of extremal rays. If R is an extremal ray, then its length l(R) is defined as follows

l(R) = min{−K_{X} · C | C is a rational curve with [C] ∈ R}.

The motivation for this definition is to give an estimate of the dimension of locus of extremal rational curves in an extremal ray. The idea of the proof of the following theorem is, basically, due to Mori [Mor1]. In the sequel we shall need the similar argument at least 4 times (!), but the details will be left to the reader.

Theorem 9.9 ([Wis]) Let C be a rational curve with its class in an extremal ray R, such that −K_{X} · C = l(R). If Q is a smooth point on C then the dimension of locus of points of curves that belong to R and pass through the point Q is at least l(R) − 1.

Proof. By the local deformation theory the space of deformations of the morphism f : ℙ^{1} → X, f(ℙ^{1}) = C, has dimension at least

h^{0}(C, f^{∗}T_{X})−h^{1}(C, f^{∗}T_{X}) = −C ·K_{X} +(1−g(C)) dim X = dim X −C ·K_{X}.

Since we want to fix the point Q, we have dim X more restrictions. Therefore, the space of deformations of the morphism f : ℙ^{1} → X fixing the point Q has dimension at least

−K_{X} · C = l(R).

The group of automorphisms of ℙ^{1} fixing a point x ∈ ℙ^{1} has dimension 2, therefore, there exists a family of rational curves on X passing through Q of dimension at least l(R) − 2. All these curves are deformations of C, therefore their numerical classes belong to R. To finish the proof it remains to verify that for a generic point P in the locus of curves from our family, there exists only finitely many curves from the family passing through P. Suppose, on the contrary, that there exists a family of rational curves passing through P and Q, parametrized by some affine curve D with smooth compactification \overline{D}. Passing to infinite points \overline{D}\D, we get limit positions of these curves, that are irreducible and reduced, since otherwise we shall obtain a contradiction with minimality of −K_{X} · C for rational curves from the ray R.

Therefore, there exists a ruled surface S and a morphism F : S → X such that C_{1} = F^{−1}(P) and C_{2} = F^{−1}(Q) are one-dimensional sections of this ruled surface over its base. Moreover, dim F (S) = 2, otherwise all curves in our pencil represent the same 1-cycle. It follows that

C_{1}^{2} < 0, C_{2}^{2} < 0, and C_{1} · C_{2} = 0,  (9.1)

the last equality is satisfied because C_{1} and C_{2} do not intersect. But this is impossible, (9.1) can not hold on a ruled surface, see e.g. [Ha1]. ⊓ ⊔

Fano varieties. A smooth variety X is called a Fano variety if its anticanonical divisor −K_{X} is ample. Its index is defined as follows:

r(X) = max{m > 0 | mH ∼ −K_{X} for some H ∈ Pic(X)}.

It is well-known that r(X) ≤ dim X + 1. Indeed, suppose that K_{X} = mH, where −H is ample, let dim X = n. The Poincare polynomial χ(vH) has at most n zeros, therefore χ(vH) ≠ 0 for some 1 ≤ v ≤ n + 1. But χ(vH) = ±h^{n}(X, vH) by the Kodaira vanishing theorem and h^{n}(X, vH) = h^{0}(X, K_{X}− vH) by the Serre duality. Therefore, m ≤ n + 1. The following theorem was conjectured by Mukai and proved in [Wis1] using the Mori theory.

Theorem 9.10 [Wis1] Let X be a n-dimensional Fano variety of index r(X). If r(X) > \frac{1}{2} dim X + 1 then Pic(X) = ℤ.

9.2 Mori Theory and Dual Varieties

9.2.A The Nef Value and the Defect

Suppose that (X, H) is a smooth polarized variety. Then the linear system |H| defines an embedding X ⊂ ℙ^{N}. Recall that X^{∗} ⊂ (ℙ^{N})^{∗} is the dual variety and

def(X, H) = codim X^{∗} − 1

is the defect of (X, H).

Theorem 9.11 If def(X, H) > 0 then K_{X} is not nef.

Proof. By Theorem 1.12 X is covered by lines. It is well-known that if a smooth projective variety is covered by rational curves then K_{X} is not nef. In our case it also follows directly from Theorem 6.19 (a). ⊓ ⊔

It follows that if def(X, H) is positive, we can apply the technique of the previous section. It turns out that the nef value and the defect of (X, H) are connected in a very simple way.

Theorem 9.12 ([BSW, BFS1]) Assume that def(X) > 0. Then

(a) The nef value τ of X is equal to

τ = \frac{dim X + def(X, H)}{2} + 1.

(b) If Sing X ∩ H, H ∈ X^{∗}, is a generic contact locus (hence a projective subspace) and l ∈ Sing X ∩ H is any line then R = ℝ_{+}[l] is an extremal ray in \overline{NE}_{1}(X) and contr_{R} coincides with the nef value morphism Φ. In particular, Φ contracts Sing X ∩ H to a point.

(c) If F is a generic fiber of the nef value morphism Φ then Pic F = ℤ. (d)

dim Φ(X) ≤ \frac{dim X − def(X, H)}{2}.

Notice that since

dim X ≡ def(X, H) mod 2

by Theorem 6.15, the nef value in this case is an integer. Theorem 9.12 can be used to show that the defect of a polarized variety is equal to 0. One needs to calculate the nef value first and then to obtain a contradiction using this Theorem. Some examples will be discussed in next sections.

Proof. We denote n = dim X and k = def(X).

Step 1. Let ℙ^{k} = Sing X ∩ H, H ∈ X^{∗}, be a generic contact locus and

l ∈ Sing X ∩ H be any line. Since (−K_{X}) · l = \frac{n + k}{2} + 1 by Theorem 6.19

(a), it follows that

τ ≥ \frac{n + k}{2} + 1.  (9.2)

Step 2. Let Φ be the nef value morphism. Since Φ is not ample, it follows from the Kleiman Criterion and the Mori Cone Theorem that there exists an extremal rational curve C contracted by Φ, (K_{X} + τH) · C = 0. We claim that C · H = 1, i.e. C is a projective line. Indeed, if C · H ≥ 2 then

τ ≤ \frac{1}{2}τH · C = \frac{1}{2}(−K_{X}) · C ≤ \frac{1}{2}(n + 1).

This contradicts (9.2). Therefore,

τ = (−K_{X}) · C = l(R),

where R = ℝ_{+}[C] is an extremal ray. Applying Theorem 9.9, wee see that if F is a positive-dimensional fiber of Φ then

dim F ≥ τ − 1.  (9.3)

Step 3. Suppose that τ > \frac{n + k}{2} + 1. Let F be a positive dimensional fiber

of Φ, x ∈ F. By Theorem 6.21 there exists a line l ⊂ X passing through x

and such that (−K_{X}) · l = \frac{n + k}{2} + 1. Therefore, Φ does not contract l, hence

l ⊄ F. Let y ∈ l, y ∉ F. Arguing as in the proof of Theorem 9.9, we see

that there exists a \frac{n + k}{2} -dimensional subvariety Y ⊂ X covered by rational

curves passing through y that are deformations of l. Applying the similar argument with ruled surfaces once again, it is easy to see that dim Y ∩F = 0. However, using (9.3) we get that

dim Y ∩ F ≥ dim Y + dim F − dim X ≥ \frac{n + k}{2} + \frac{n + k}{2} − n > 0.

Contradiction. This proves (a). Step 4. Clearly, Φ contracts any generic contact locus Sing X ∩ H, H ∈ X^{∗}. Moreover, arguing as above (or using Theorem 6.14 (d)), we see that any fiber

of Φ is at least \frac{n + k}{2} -dimensional, therefore dim Y ≤ n − \frac{n + k}{2} = \frac{n − k}{2} .

This proves (d). Let F be a generic fiber. Then K_{F} = K_{X}|_{F} = −τH|_{F}. Therefore, F is a Fano variety of index at least

τ = \frac{n + k}{2} + 1 > \frac{dim F}{2} + 1.

Therefore, Pic F = ℤ by Theorem 9.10. This proves (c).

Step 5. It remains to prove that Φ is a contraction of an extremal ray ℝ_{+}[l]. Suppose, on the contrary, that there exists an extremal ray R such that l ∉ R and Φ contracts any curve from R. Let C be an extremal rational curve from R. Arguing as in Step 2, we see that l(R) = (−K_{X}) · C = τ. Consider the morphism Ψ = contr_{R}. By Theorem 9.9, there exists a fiber F of Ψ such that dim F ≥ τ − 1, let x ∈ F. By Theorem 6.21 there exists a line l ⊂ X passing through x and such that (−K_{X}) · l = τ. This line is either a line in a generic contact locus or its limit position, therefore Ψ does not contract l. Hence l ⊄ F. Let y ∈ l, y ∉ F. Arguing as in the proof of Theorem 9.9, we see that there exists a (τ − 1)-dimensional subvariety Y ⊂ X covered by rational curves passing through y that are deformations of l. Applying the similar argument once again, it is easy to see that dim Y ∩ F = 0. But

dim Y ∩ F ≥ dim Y + dim F − dim X ≥ 2τ − 2 − n = k > 0.

Contradiction. ⊓ ⊔

9.2.B The Defect of Fibers of the Nef Value Morphism

Suppose now that (X, H) is a smooth polarized variety and K_{X} is not nef. Let Φ : X → Y denote a nef value morphism. Then its generic fiber F is a smooth irreducible variety. It has a canonical polarization (F, H_{F}), where H_{F} is a hyperplane section of F in the embedding F ⊂ X ⊂ ℙ^{N}.

Theorem 9.13 ([BFS1]) The following conditions are equivalent:

(i) def(X, H) > 0; (ii) def(F, H_{F}) > dim Y; (iii) def(X, H) = def(F, H_{F}) − dim Y > 0.

Proof. It follows from Theorem 6.24 that

dim X + def(X, H) ≥ dim F + def(F, H_{F}).

Therefore, (i) follows from (ii). Obviously (ii) follows from (iii). Therefore, it is sufficient to prove that (iii) follows from (i).

Since def(X, H) > 0 we have

τ(X, H) = \frac{dim X + def(X, H)}{2} + 1

by Theorem 9.12. Since F is a generic fiber of Φ we have

O(K_{F} + τH_{F}) = O(K_{X} + τH)|_{F}

is a trivial bundle. Therefore, K_{F} is not nef and τ(F, H_{F}) = τ(X, H). If def(F, H_{F}) > 0 then again by Theorem 9.12

def(F, H_{F}) = \frac{dim F + def(F, H_{F})}{2} + 1

and, therefore,

0 < def(X, H) = − dim X + dim F + def(F, H_{F}) = def(F, H_{F}) − dim Y.

It remains to show that if def(X, H) > 0 then def(F, H_{F}) > 0 as well. Let L = O(H), L_{F} = O(H_{F}). Recall that for any line bundle L on a smooth variety X we denote by J_{1}(X, L) the vector bundle of 1-jets of L; for any vector bundle V we denote by c_{r}(V) its r Chern class. By Theorem 7.1 we need to show that c_{dim F} (J_{1}(F, L_{F})) = 0. From the exact sequence

0 → T_{F}^{∗}(X) ⊗ L_{F} → J_{1}(X, L)|_{F} → J_{1}(F, L_{F}) → 0

we see that the total Chern class

c(J_{1}(X, L)|_{F}) = c(T_{F}^{∗}(X) ⊗ L_{F}) · c(J_{1}(F, L_{F})).

Since F is a generic fiber of Φ, T_{F}^{∗}(X) ⊗ L_{F} = L_{F} + … + L_{F} (dim Y copies). Therefore, c(T_{F}^{∗}(X) ⊗ L_{F}) = (1 + H_{F})^{dim Y}. In particular,

c_{dim F} (J_{1}(X, L)|_{F}) =

c_{dim F} (J_{1}(F, L_{F})) + ∑_{i=1}^{dim Y} \binom{dim Y}{i}c_{dim F −i}(J_{1}(F, L_{F})) · H_{F}^{i} .

All summands on the right are nonnegative since H_{F} is very ample and J_{1}(F, L_{F}) is spanned. Therefore in order to prove that c_{dim F} (J_{1}(F, L_{F})) = 0 it remains to prove that c_{dim F} (J_{1}(X, L)|_{F}) = 0.

It will be easier to prove a more general statement that c_{k}(J_{1}(X, L)|_{F}) = 0 for k ≥ dim F −def(X, H)+1. By the classical representation of a Chern class of a spanned vector bundle [Fu1] we see that c_{k}(J_{1}(X, L)) = 0 is represented by a set C_{k} ⊂ X of all points where dim X − k + 2 generic sections of H^{0}(X, J_{1}(X, L)) are not linearly independent. We may suppose that these sections have a form j_{1}(s_{1}), …, j_{1}(s_{dim X−k+2}), where s_{1}, …, s_{dim X−k+2} are generic sections of H^{0}(X, L) and j_{1} is a natural map sending a germ of a section to its 1-jet. It suffices to prove that for k ≥ dim F − def(X, H) + 1 we have C_{k} ∩ F = ∅. Since C_{k} ⊃ C_{k+1}, we only need to prove that C ∩ F = ∅, where C = C_{dim F −def(X,H)+1}. Suppose that this intersection is non-empty. Therefore, the restriction map Φ_{C} : C → Y is surjective. Since dim C = dim Y + def(X, H) − 1, it follows that a generic fiber of Φ_{C} has dimension def(X, H) − 1.

Suppose that x ∈ C ∩F. There exist λ_{i} for 1 ≤ i ≤ dim Y +def(X, H)−1, not all zero, such that ∑_{i} λ_{i}j_{1}(s_{i}(x)) = 0. Therefore s = ∑_{i} λ_{i}s_{i} is a nontrivial global section of L vanishing at x with its 1-jet. Hence the divisor H_{s} is singular at x. By Theorem 6.21 there exists a linear subspace ℙ^{def(X,H)} ⊂ X containing x, contained in Sing H_{s}, and such that O(K_{X}+τ(X, H)H)|_{ℙ^{def(X,H)}} is a trivial bundle. Since this is a trivial bundle, ℙ^{def(X,H)} is contracted by Φ to a point and, therefore, ℙ^{def(X,H)} ⊂ F. On the other hand, since ℙ^{def(X,H)} is contained in Sing H_{s} it is also contained in C. Therefore, a generic fiber of Φ_{C} has dimension greater or equal to def(X, H). Contradiction. ⊓ ⊔

9.2.C Varieties With Small Dual Varieties

In this section we describe without proofs smooth polarized varieties (X, H) with positive defect for dim X ≤ 10. First we need a couple of definitions.

Definition 9.14 A smooth polarized variety (X, H) is called a ℙ^{d}-bundle over a normal variety Y if there exists a surjective morphism p : X → Y such that all the fibers F of p are projective subspaces ℙ^{d} of ℙ^{N}, where X ⊂ ℙ^{N} is an embedding corresponding to H.

Definition 9.15 A smooth polarized variety (X, H) is called a scroll (resp. a quadric, Del Pezzo, or Mukai fibration) over a normal variety Y if there exists a surjective morphism p : X → Y with connected fibers such that K_{X} +(dim X −dim Y +1)H ∼ p^{∗}H′ (resp. K_{X} +(dim X −dim Y)H ∼ p^{∗}H′, K_{X} +(dim X −dim Y −1)H ∼ p^{∗}H′, or K_{X} +(dim X −dim Y −2)H ∼ p^{∗}H′) for some ample Cartier divisor H′ on Y.

To adjust this definition we need to note that it easily follows from the Kobayashi–Ochiai characterisation of projective spaces and quadrics [KO] that if (X, H) is a scroll over Y then a generic fiber F of p is a projective subspace ℙ^{d} of ℙ^{N}, where X ⊂ ℙ^{N} is an embedding corresponding to H. If (X, H) is a quadric fibration then a generic fiber F is isomorphic to a quadric hypersurface Q in some projective space and the line bundle corresponding to this embedding is O(H)|_{F}. In fact, quadric fibrations will not appear in the classification because if (X, H) is a quadric fibration then def(X, H) = 0 by [BFS1].

dim X ≤ 2. If dim X ≤ 2 then def(X, H) = 0.

dim X = 3. Suppose that dim X = 3. Then def(X, H) > 0 if an only if (X, H) is a ℙ^{2}-bundle over a smooth curve. In this case def(X, H) = 1.

dim X = 4. Suppose that dim X = 4. Then def(X, H) > 0 if an only if (X, H) is a ℙ^{3}-bundle over a smooth curve. In this case def(X, H) = 2.

dim X = 5. Let dim X = 5 and def(X, H) > 0. Then we have the following possibilities:

(a) X is a hyperplane section of the Plücker embedding of the Grassmanian Gr(2, 5) in ℙ^{9}, def(X, H) = 1.

(b) (X, H) is a ℙ^{4}-bundle over a smooth curve, def(X, H) = 3.

(c) (X, H) is a ℙ^{3}-bundle over a smooth surface, def(X, H) = 1. dim X = 6. Let dim X = 6 and def(X, H) > 0. Then either:

(a) (X, H) is the Plücker embedding of the Grassmanian Gr(2, 5) in ℙ^{9}, def(X, H) = 2.

(b) (X, H) is a ℙ^{5}-bundle over a smooth curve, def(X, H) = 4.

(c) (X, H) is a ℙ^{4}-bundle over a smooth surface, def(X, H) = 2.

dim X = 7. Let dim X = 7 and def(X, H) > 0. Then we have the following possibilities:

(a) (X, H) is a ℙ^{6}-bundle over a smooth curve, def(X, H) = 5.

(b) (X, H) is a ℙ^{5}-bundle over a smooth surface, def(X, H) = 3.

(c) X is a 7-dimensional Fano variety with Pic X = ℤ and K_{X} ∼ −5H. In this case def(X, H) = 1. An example of such variety is the section of 10-dimensional spinor variety S in ℙ^{15} by three generic hyperplanes [Mu, LS].

(d) (X, H) is a Del Pezzo fibration over a smooth curve such that the general fiber F is isomorphic to Gr(2, 5) and the embedding corresponding to O(H)|_{F} is the Plücker embedding in ℙ^{9}. In this case def(X, H) = 1.

(e) (X, H) is a scroll over a 3-dimensional variety, def(X, H) = 1.

dim X = 8. Let dim X = 8 and def(X, H) > 0. Then either:

(a) (X, H) is a ℙ^{7}-bundle over a smooth curve, def(X, H) = 6.

(b) (X, H) is a ℙ^{6}-bundle over a smooth surface, def(X, H) = 4.

(c) X is is a 8-dimensional Fano variety with Pic X = ℤ and K_{X} ∼ −6H. In this case def(X, H) = 2.

(d) (X, H) is a scroll over a 3-dimensional variety, def(X, H) = 2.

dim X = 9. Let dim X = 9 and def(X, H) > 0. Then either:

(a) (X, H) is a ℙ^{8}-bundle over a smooth curve, def(X, H) = 7.

(b) (X, H) is a ℙ^{7}-bundle over a smooth surface, def(X, H) = 5.

(c) X is a 9-dimensional Fano variety with Pic X = ℤ and K_{X} ∼ −7H. In this case def(X, H) = 3.

(d) (X, H) is a scroll over a 3-dimensional variety, def(X, H) = 3.

(e) X is a Fano 9-dimensional variety with K_{X} ∼ −6H and and Pic X = ℤ, def(X, H) = 1.

(f) (X, H) is a Mukai fibration over a smooth curve with Pic F = ℤ for a generic fiber F, def(X, H) = 1.

(g) (X, H) is a scroll over a 4-dimensional variety, def(X, H) = 1.

dim X = 10. Let dim X = 10 and def(X, H) > 0. Then we have the following possibilities:

(a) (X, H) is a ℙ^{9}-bundle over a smooth curve, def(X, H) = 8.

(b) (X, H) is a ℙ^{8}-bundle over a smooth surface, def(X, H) = 6.

(c) X is a 10-dimensional Fano variety with Pic X = ℤ and K_{X} ∼ −8H. In this case def(X, H) = 4.

(d) (X, H) is a scroll over a 3-dimensional variety, def(X, H) = 4.

(e) X is a Fano 10-dimensional variety with K_{X} ∼ −7H and Pic X = ℤ, def(X, H) = 2;

(f) (X, H) is a Mukai fibration over a smooth curve with Pic F = ℤ for a generic fiber F, def(X, H) = 2.

(g) (X, H) is a scroll over a 4-dimensional variety, def(X, H) = 2.

9.3 Polarized Flag Varieties With Positive Defect

9.3.A Nef Value of Polarized Flag Varieties

We continue to use the notation from Section 2.1. Let G be a connected simply-connected semisimple complex algebraic group with a Borel subgroup B and a maximal torus T ⊂ B. Let P be a character group of T (the weight lattice). Let Δ ⊂ P be a set of roots of G relative to T. To every root α ∈ Δ we can assign the 1-dimensional unipotent subgroup U_{α} ⊂ G. We define the negative roots Δ^{−} as those roots α such that U_{α} ⊂ B. Let Π ⊂ Δ^{+} be simple roots, Π = {α_{1}, …, α_{n}}, where n = rank G = dim T. The root lattice Q ⊂ P is a sublattice spanned by Δ. For any simple root α_{i} ∈ Π we can associate a smooth curve C_{α_{i}} ⊂ G/B, namely C_{α_{i}} = \overline{π(U_{α_{i}})}, where π : G → G/B is a factorization map. By linearity to any element α = ∑ n_{i}α_{i} ∈ Q we assign a 1-cycle C_{α} = ∑ n_{i}C_{α_{i}} in G/B. Clearly this cycle is effective if α ∈ Q^{+}, the semigroup generated by simple roots.

P is generated as a ℤ-module by fundamental weights ω_{1}, …, ω_{n} dual to the simple roots under the Killing form. We denote the dominant weights by P^{+} and the strictly dominant weights by P^{++}.

The character group of B is identified with a character group of T. Therefore, for any λ ∈ P we can assign a 1-dimensional B-module ℂ_{λ}, where B acts on ℂ_{λ} by a character λ. Now we can define the twisted product G×_{B} ℂ_{λ} to be the quotient of G × E by the diagonal action B:

b · (g, z) = (gb^{−1}, b · z).

Projection onto the first factor induces a map G×_{B}ℂ_{λ} → G/B, which realizes the twisted product as an equivariant line bundle L_{λ} on G/B with fiber ℂ_{λ}.

The following theorem is well-known:

Theorem 9.16

(1) The correspondence α → C_{α} is an isomorphism of Q and N_{1}^{ℤ}(G/B), the group of 1-cycles modulo numerical equivalence.

(2) The correspondence λ → L_{λ} is an isomorphism of P and Pic(G/B). The latter group is, in turn, isomorphic to N_{Z}^{1}(G/B), the group of Cartier divisors modulo numerical equivalence.

(3) The pairing P × Q → ℤ given by duality corresponds to the intersection product on N_{Z}^{1}(G/B) × N_{1}^{Z}(G/B).

(4) L_{λ} is nef if and only if λ ∈ P^{+}. L_{λ} is ample if and only if L_{λ} is very ample if and only if λ ∈ P^{++}.

From this theorem it is easy to derive the corresponding description for flag varieties G/P, where P ⊂ G is an arbitrary parabolic subgroup. Let Π_{P} ⊂ Π be some subset of simple roots. Let Δ_{P}^{+} ⊂ Δ^{+} denote the positive roots that are linear combinations of the roots in Π_{P}. Then we may suppose that P is generated by B and by the root groups U_{α} for α ∈ Δ_{P}^{+}. We denote Π \ Π_{P} by Π_{G/P} and Δ^{+} \ Δ_{P}^{+} by Δ_{G/P}^{+} . A parabolic subgroup is maximal if and only if Π_{G/P} is a single simple root. We denote by Q_{G/P} the sublattice of Q generated by Π_{G/P} and by Q_{G/P}^{+} the corresponding subsemigroup. For any α_{i} ∈ Π_{G/P} we can associate a smooth curve C_{α_{i}} ⊂ G/P, namely C_{α_{i}} = \overline{π(U_{α_{i}})}, where π : G → G/P is a factorization map. By linearity to any element α = ∑ n_{i}α_{i} ∈ Q_{G/P} we assign a 1-cycle C_{α} = ∑ n_{i}C_{α_{i}} in G/P. Clearly this cycle is effective if α ∈ Q_{G/P}^{+} .

The fundamental weights ω_{i_{1}}, … ω_{i_{k}} dual to the simple roots in Π_{G/P} generate the sublattice P_{G/P} of P. We denote P^{+} ∩ P_{G/P} by P_{G/P}^{+} . The subset P_{G/P}^{++} ⊂ P_{G/P}^{+} consists of all weights λ = ∑ n_{k}ω_{i_{k}} such that all n_{k} > 0. Any weight λ ∈ P_{G/P} defines a character of P, and, therefore, a line bundle L_{λ} on G/P.

Theorem 9.17

(1) The correspondence α → C_{α} is an isomorphism of Q_{G/P} and N_{1}^{ℤ}(G/P).

(2) The correspondence λ → L_{α} is an isomorphism of P_{G/P} and Pic(G/P). The latter group is, in turn, isomorphic to N_{Z}^{1}(G/P). In particular, Pic(G/P) = ℤ if and only if P is maximal.

(3) The pairing P_{G/P} × Q_{G/P} → ℤ given by duality corresponds to the intersection product on N_{Z}^{1}(G/P) × N_{1}^{Z}(G/P).

(4) L_{λ} is nef if and only if λ ∈ P_{G/P}^{+} . L_{λ} is ample if and only if L_{λ} is very ample if and only if λ ∈ P_{G/P}^{++} .

(5) If λ ∈ P_{G/P}^{+} then the linear system corresponding to L_{λ} is base–point free. The corresponding map given by sections is a factorization G/P → G/Q, where Q is a parabolic subgroup such that Π_{Q} is a union of Π_{P} and all simple roots in Π_{G/P} orthogonal to λ.

Now it is quite straightforward to find nef values of ample line bundles.

Theorem 9.18

(1) The anticanonical bundle K_{G/P}^{∗} corresponds to the dominant weight ρ_{G/P} = ∑_{α∈Δ_{G/P}^{+}} α, ρ_{G/P} ∈ P_{G/P}^{++} . Canonical line bundle K_{G/P} is not nef.

(2) The nef value of an ample line bundle L_{λ} given by the weight λ ∈ P_{G/P}^{++} is equal to

τ = max_{α∈Π_{G/P}} \frac{(ρ_{G/P} , α)}{(λ, α)}.

(3) The nef value morphism Φ is a homogeneous fiber bundle Φ : G/P → G/Q, where Q is the parabolic subgroup defined by Π_{Q} = Π_{P} ∪Π′, where Π′ ⊂ Π_{G/P} is a set of simple roots for which the above maximum occurs.

Proof. The tangent bundle T (G/P) is an equivariant vector bundle such that its fiber over the identity coset is isomorphic to the quotient of Lie algebras Lie G/ Lie P on which P acts via an adjoint representation on Lie G projected to the quotient. The T -weights of this representation consist of Δ_{G/P}^{+} . Therefore the anticanonical bundle K_{G/P}^{∗} = Λ^{dim G/P} T (G/P) corresponds to the weight ρ_{G/P}. Notice that for any α ∈ Π_{G/P}

(ρ_{G/P} , α) = ( ∑_{β∈Δ^{+}} β, α) − ( ∑_{β∈Δ_{P}^{+}} β, α) .

The first product is equal to 2, the second one is non-positive, because any root from Δ_{P}^{+} is a linear combination of simple roots from Π_{P} with nonnegative coefficients and the scalar product of two distinct simple roots is non-positive. Therefore, ρ_{G/P} ∈ P_{G/P}^{++} and K_{G/P} is not nef.

Let

ρ_{G/P} = ∑_{α_{i}∈Π_{G/P}} n_{i}ω_{i}, λ_{G/P} = ∑_{α_{i}∈Π_{G/P}} m_{i}ω_{i}.

Then, clearly, the nef value τ of L is given by

τ = max_{α_{i}∈Π_{G/P}} \frac{n_{i}}{m_{i}} = max_{α∈Π_{G/P}} \frac{(ρ_{G/P} , α)}{(λ, α)}.

Part (3) follows from Theorem 9.17, part (5). ⊓ ⊔ Using this theorem, it is straightforward to calculate nef values in par-

ticular cases. Suppose that G is a simple group, P is a maximal parabolic subgroup such that Π_{G/P} = {α_{i}} is a single simple root. Suppose further that L is a generator of Pic G/P, that is, L corresponds to the fundamental weight ω_{i}. In the following tables we give for each G and i (in the Bourbaki numbering of fundamental weights) the dimension dim G/P and the nef value τ of L.

G = A_{l}  \begin{matrix}i & 1, …, l \\ dim G/P & i(l + 1 − i) \\ τ & l + 1 \end{matrix}

G = B_{l}  \begin{matrix}i & 1, …, l − 1 & l \\ dim G/P & i(4l + 1 − 3i)/2 & l(l + 1)/2 \\ τ & 2l − i & 2l \end{matrix}

G = C_{l}  \begin{matrix}i & 1, …, l \\ dim G/P & i(4l + 1 − 3i)/2 \\ τ & 2l − i + 1 \end{matrix}

G = D_{l}  \begin{matrix}i & 1, …, l − 2 & l − 1, l \\ dim G/P & i(4l − 1 − 3i)/2 & l(l − 1)/2 \\ τ & 2l − i − 1 & 2l − 2 \end{matrix}

G = E_{6}  \begin{matrix}i & 1, 5 & 2, 4 & 3 & 6 \\ dim G/P & 16 & 25 & 29 & 21 \\ τ & 12 & 9 & 7 & 11 \end{matrix}

G = E_{7}  \begin{matrix}i & 1 & 2 & 3 & 4 & 5 & 6 & 7 \\ dim G/P & 27 & 42 & 50 & 53 & 47 & 33 & 42 \\ τ & 18 & 13 & 10 & 8 & 11 & 17 & 14 \end{matrix}

G = E_{8}  \begin{matrix}i & 1 & 2 & 3 & 4 & 5 & 6 & 7 & 8 \\ dim G/P & 57 & 83 & 97 & 104 & 106 & 98 & 78 & 92 \\ τ & 29 & 19 & 14 & 11 & 9 & 13 & 23 & 17 \end{matrix}

G = F_{4}  \begin{matrix}i & 1 & 2 & 3 & 4 \\ dim G/P & 15 & 20 & 20 & 15 \\ τ & 8 & 5 & 7 & 11 \end{matrix}

G = G_{2}  \begin{matrix}i & 1 & 2 \\ dim G/P & 5 & 5 \\ τ & 5 & 3 \end{matrix}

9.3.B Classification

In this section we use the calculations from the previous section and Theorems 9.12, 9.13 to classify all polarized flag varieties with positive defect. We continue to use notations from the previous section.

Suppose first that G is a simple group, P ⊂ G is a maximal parabolic subgroup corresponding to a simple root α_{i}. Then all very ample linear bundles on G/P have a form L^{⊗k} for k > 0, where L corresponds to the fundamental weight ω_{i}. If k > 1 then by Theorem 7.15 we have def(G/P, L^{⊗k}) = 0. Therefore it suffices to find def(G/P, L). By Theorem 9.12 if def(G/P, L) > 0 the def(G/P, L) = 2τ(G/P, L) − 2 − dim G/P. Therefore we may look through the table of the previous section and pick all entries such that d_{P} = 2τ(G/P, L) − 2 − dim G/P > 0. The following table contains all these entries. Therefore all polarized flag varieties with positive defect (such that P is a maximal parabolic subgroup) are contained in this table.

No.  G  i  τ(G/P, L) dim G/P  d_{P}
1  A_{l}  1, l  l + 1  l  l
2 A_{l}, l even 2, l − 1  l + 1  2l − 2  2
3  C_{l}  1  2l  2l − 1 2l − 1
4  B_{2}  2  4  3  3
5  B_{4}  4  8  10  4
6  D_{5}  4, 5  8  10  4
7 A_{l}, l odd 2, l − 1  l + 1  2l − 2  2
8  A_{5}  3  6  9  1
9  B_{l}  1  2l − 1  2l − 1 2l − 3
10  B_{3}  3  6  6  4
11  D_{4}  3, 4  6  6  4
12  B_{5}  5  10  15  3
13  D_{6}  5, 6  10  15  3
14  B_{6}  6  12  21  1
15  D_{7}  6, 7  12  21  1
16  C_{l}  2  2l − 1  4l − 5  1
17  D_{l}  1  2l − 2  2l − 2 2l − 4
18  E_{6}  1, 5  12  16  6
19  E_{7}  1  18  27  7
20  F_{4}  4  11  15  5
21  G_{2}  1  5  5  3

In fact, only first 6 entries of this table correspond to polarized flag varieties with positive defect, and entry 4 is a particular case of entry 3. This can be done case-by-case (see [Sn]). Therefore, we have the following theorem

Theorem 9.19 Suppose that P_{i} is a maximal parabolic subgroup of a simple algebraic group corresponding to the simple root α_{i}. Let L be a very ample line bundle on G/P_{i} corresponding to the fundamental weight ω_{i}. Then

(1) def(G/P_{i}, L^{⊗k}) = 0 for k > 1.

(2) def(G/P_{i}, L) > 0 if and only if G/P_{i} is one of the following:

– A_{l}/P_{1}, A_{l}/P_{l}. def(G/P) = dim(G/P) = l.

– A_{l}/P_{2}, A_{l}/P_{l−1}, l ≥ 4 even. def(G/P) = 2, dim(G/P) = 2l − 2.

– C_{l}/P_{1}. def(G/P) = dim(G/P) = 2l − 1.

– B_{4}/P_{4}, D_{5}/P_{4}, D_{5}/B_{5}. def(G/P) = 4, dim(G/P) = 10.

Notice that we do not follow the usual tradition and consider isomorphic polarized varieties with different group actions as distinct varieties. The next step is to consider flag varieties corresponding to arbitrary parabolic subgroups in simple group.

Theorem 9.20 Suppose that (G/P, L) is a polarized flag variety of a simple algebraic group G, where P ⊂ G is a non-maximal parabolic subgroup. Then def(G/P, L) = 0.

Proof. Suppose that def(G/P, L) > 0. By Theorem 9.18 the nef value morphism is a homogeneous fiber bundle Φ : G/P → G/Q, where Q ⊂ G is another parabolic subgroup. The fiber F of Φ is isomorphic to Q/P.

By Theorem 9.12 we have Pic F = ℤ, therefore F is isomorphic to a quotient of a simple group by a maximal parabolic subgroup and Q is a proper subgroup of G. By Theorem 9.13 def(F, L) > dim G/Q > 0. Therefore, the polarized flag variety (F, L) is one of the varieties from Theorem 9.19. Let us consider 3 cases.

If F is equal to A_{l}/P_{2} or A_{l}/P_{l−1}, l ≥ 4 even, then def(F, L) = 2. Therefore, dim G/Q = 1. It follows that G = SL_{2}, Q is a Borel subgroup, and there are no possibilities for P.

If F is equal to B_{4}/P_{4}, D_{5}/P_{4}, or D_{5}/B_{5} then def(F, L) = 4. Therefore, dim G/Q is equal to 1, 2, or 3. It follows that G is equal to SL_{2}, SL_{3}, SL_{4}, or Sp_{4}. Again none of these groups contain B_{4} or D_{5} as a simple component of a Levi subgroup.

Finally, let F be A_{l}/P_{1}, A_{l}/P_{l}, or C_{\frac{l+1}{2}} /P_{1}, so F is isomorphic to ℙ^{l}. We want to show that dim G/Q > k, which will be a contradiction. The set of simple roots Π_{Q} contains a subset Ψ with Dynkin diagram of type A_{l} or C_{\frac{l+1}{2}} . Moreover, there exists a simple root α ∈ Π such that α ∉ Π_{Q} and the Dynkin diagram of Ψ′ = Ψ ∪{α} is connected. Then Ψ′ is a set of simple roots of a simple subgroup G′ ⊂ G and dim G/Q ≥ dim G′/Q′, where Q′ = G′ ∩ Q is a maximal parabolic subgroup defined by Ψ. It is easy to check that in all arising cases dim G′/Q′ > k. ⊓ ⊔

Now we can handle the case of an arbitrary semisimple group and its arbitrary parabolic subgroup.

Theorem 9.21 Suppose that (G/P, L) is a polarized flag variety of a complex semisimple algebraic group G. Then def(G/P, L) > 0 if and only if either (G/P, L) is one of polarized varieties described in Theorem 9.19 or the following conditions hold:

– G = G_{1} × G_{2}, P = P_{1} × P_{2}, where P_{1} ⊂ G_{1}, P_{2} ⊂ G_{2}.

– L = pr_{1}^{∗} L_{1} ⊗ pr_{2}^{∗} L_{2}, where pr_{i} : G/P → G_{i}/P_{i} are projections and L_{i} are very ample line bundles on G_{i}/P_{i}.

– (G_{1}/P_{1}, L_{1}) is one of polarized varieties described in Theorem 9.19.

– def(G_{1}/P_{1}, L_{1}) > dim G_{2}/P_{2}.

In this case def(G/P, L) = def(G_{1}/P_{1}, L_{1}) − dim G_{2}/P_{2}.

Proof. This Theorem easily follows from Theorem 6.5. ⊓ ⊔

150 9. Mori Theory and Dual Varieties

10. Some Applications of the Duality

Preliminaries

In this chapter we collect several applications, variations, and illustrations of the projective duality.

10.1 Discriminants and Automorphisms

10.1.A Matsumura–Monsky Theorem

Let D ⊂ ℙ^{n} be a smooth hypersurface of degree d. It was first proved in [MM] that the group of projective automorphisms preserving D is finite if d > 2. In fact, it was also proved that the group of biregular automorphisms of D is finite if d > 2 (except the cases d = 3, n = 2 and d = 4, n = 3). Though this generalization looks much stronger, actually it is an easy consequence of the “projective” version and the Bart Theorem [Ba].

More generally, let G/P be a flag variety of a simple Lie group and D ⊂ G/P be a smooth ample divisor. Let L_{λ} = O(D) be the corresponding ample line bundle, where λ ∈ P^{+} is a dominant weight. Then one might expect that the normalizer N_{G}(D) of D in G is finite if λ is big enough. Indeed, if N_{G}(D) (or actually any linear algebraic group of transformations of D) contains a one-parameter subgroup of automorphisms of D then D is covered by rational curves. However, if λ is big enough then the canonical class K_{D} is nef by the adjunction formula, and, therefore, D can not be covered by rational curves. Unfortunately, this transparent approach does not give strong estimates on λ. Much better estimates can be obtained using an original proof of Matsumura-Monsky theorem.

The problem can be reformulated as follows. Suppose that V_{λ} is an irreducible G-module with highest weight λ. Let D ⊂ V_{λ} be the discriminant variety (the dual variety to the orbit of the highest weight vector in V_{λ}^{∗}, see Section 9.3 for further details). We shall show that if λ is big enough then any point x ∈ V_{λ} \ D has a finite stabilizer G_{x} and the orbit of x is closed, Gx = \overline{Gx} (therefore x is a stable point of V_{λ} in the sense of Geometric Invariant Theory, see [MFK]). This result can be compared with the results of [AVE], where all irreducible modules of simple algebraic groups with infinite stabilizers of generic points were found (this classification was extended later in [El1] and [El2] to handle irreducible representations of semisimple groups and any representations of simple groups as well).

If D is not a hypersurface then an easy inspection using Theorem 9.21 shows that the stabilizer of any point is infinite. So from now on we shall assume that D is a hypersurface defined by vanishing of the discriminant Δ.

Recall that a dominant weight λ is called self-dual if V_{λ} is isomorphic to V_{λ}^{∗} as a G-module. Let P_{S}^{+} ⊂ P^{+} be the subcone of self-dual dominant weights. Let γ be the highest root.

Theorem 10.1 Let V_{λ} be an irreducible representation of a simple algebraic group G with highest weight λ such that D is a hypersurface. Suppose that (λ − γ, μ) > 0 for any μ ∈ P_{S}^{+}. Let x ∈ V_{λ} \ D. Then G_{x} is finite and Gx = \overline{Gx}. Moreover, G_{[x]} is also finite, where [x] is the line spanned by x.

Proof. If G_{x} is finite and G_{[x]} is infinite then [x] \ {0} ⊂ Gx. Therefore, 0 ∈ \overline{Gx} and Δ(x) = Δ(0) = 0, hence, x ∈ D. The same argument shows that if x ∈ V_{λ} \ D then \overline{Gx} ⊂ V_{λ} \ D. If Gx is not closed then G_{y} is infinite for any point y ∈ \overline{Gx} \ Gx. Therefore, in order to prove the Theorem it suffices to prove that for any x ∈ V_{λ} \ D the stabilizer G_{x} is finite.

Suppose that G_{x} is infinite. Then G_{y} is infinite and reductive (see e.g. [PV]) for any point y from the closed orbit in \overline{Gx}. Therefore, it suffices to prove that if S ⊂ G is a one-dimensional torus and Sx = x then x ∈ D.

Without loss of generality we may assume that S ⊂ T, where T is the fixed maximal torus, and 𝔱 = Lie T is the Cartan subalgebra. Let x = ∑_{π∈P} x_{π} be the weight decomposition of x. Let Supp(x) = {π ∈ P | x_{π} ≠ 0} be the support of x. For any μ ∈ P let H_{μ} denote the hyperplane of weights perpendicular to μ. Then there exists μ ∈ P such that Supp(x) ⊂ H_{μ}. Using the action of the Weil group we may assume that μ ∈ P^{+}.

For any λ ∈ P^{+} let λ^{∗} be the highest weight of the dual module V_{λ}^{∗}. Then λ^{∗} = −w_{0}(λ), where w_{0} is the longest element of the Weil group.

Suppose that (μ, λ^{∗} − γ) > 0. Then for any positive root α we have (μ, λ^{∗} − α) > 0. Therefore, x is perpendicular to [𝔤, v_{λ^{∗}}], where v_{λ^{∗}} is the highest weight vector of V_{λ}^{∗}. It follows that x ∈ D.

Suppose that (μ^{∗}, λ^{∗} − γ) > 0. Then for any positive root α we have (μ^{∗}, λ^{∗} − α) > 0. Therefore, w_{0}(x) is perpendicular to [𝔤, v_{λ^{∗}}]. It follows that w_{0}(x) ∈ D. Therefore, It follows that x ∈ D as well.

Suppose now that (μ, λ^{∗} − γ) ≤ 0 and (μ^{∗}, λ^{∗} − γ) ≤ 0. Then

(μ + μ^{∗}, λ^{∗} − γ) ≤ 0.

But μ + μ^{∗} is a self-dual weight, hence this contradicts assumptions of the Theorem. ⊓ ⊔ Example 10.2 If G = SL_{n} and λ = ∑ n_{i}ω_{i}, where ω_{1}, …, ω_{n} are the fundamental weights, then the assumptions of the Theorem are satisfied if and only if ∑ n_{i} > 2, for example if λ = nω_{1}, n > 2. In particular, we recover the original Matsumura–Monsky Theorem.

10.1.B Quasiderivations of Commutative Algebras

Commutative algebras without identities. Let V = ℂ^{n}. Consider the vector space A = S^{2}V^{∗} ⊗ V parametrizing bilinear commutative multiplications in V. In the sequel we identify points of A with the corresponding commutative algebras.

Definition 10.3 Let A ∈ A. A non-zero element v ∈ A is called a quadratic nilpotent if v^{2} = 0. Let D_{1} ⊂ A be a subset of all algebras containing quadratic nilpotents. A one-dimensional subalgebra U ⊂ A is called singular if there exists linear independent vectors u ∈ U and v ∈ A such that

u^{2} = αu, uv = \frac{α}{2}v, where α ∈ ℂ.

Let D_{2} ⊂ A be a subset of all algebras containing singular subalgebras.

Then the following theorem holds.

Theorem 10.4 ([T6])

(1) D_{1} and D_{2} are irreducible hypersurfaces.

(2) Let A ∈ A. Then A contains a one-dimensional subalgebra.

(3) Let A ∈ A \ (D_{1} ∪ D_{2}). Then A contains exactly 2^{n} − 1 one-dimensional subalgebras; all these subalgebras are spanned by idempotents.

Proof. Clearly A can be identified with a set of at most n-dimensional linear systems of quadrics in V. Namely, any linear function f ∈ V^{∗} defines a homogeneous quadratic function

v → f(v^{2}).

Then D_{1} corresponds to the set of linear systems of quadrics with zero resultant. This proves that D_{1} is an irreducible hypersurface.

There exists, however, another useful identification. Any algebra A determines the n-dimensional linear system of affine quadrics in V. Namely, any linear form f ∈ V^{∗} defines an affine quadratic form

v → f(v − v^{2}).

We can embed V into a projective space ℙ as an affine chart. Then this linear system is naturally identified with a n-dimensional linear system of quadrics in ℙ. Clearly, the base points of this linear system that do not lie on the infinity coincide with idempotents of A. Infinite base points are the projectivizations of lines spanned by quadratic nilpotents. It is easy to see that quadrics from our linear system intersect transversally at 0. Moreover, quadrics intersect non-transversally at some point v ≠ 0 if and only if the subalgebra spanned by v (if v is finite) or the subalgebra with the projectivization v (if v is infinite) is singular. Therefore, (3) follows from the Bezout theorem.

D_{2} is irreducible since D_{2} = GL_{n} ·D′_{2}, where D′_{2} ⊂ A is the linear subset of all algebras that have a fixed singular subalgebra and a fixed line spanned by the vector v from the definition of a singular subalgebra. It is also quite easy to check that D_{2} is actually a closed hypersurface.

Since quadrics of our linear system intersect transversally at 0, it follows that there exist other base points, i.e. there exists at least one 1-dimensional subalgebra. ⊓ ⊔

The algebras A ∈ A that do not belong to discriminant varieties D_{1} and D_{2} will be called regular. Of course, both hypersurfaces D_{1} and D_{2} can be interpreted as ordinary discriminants. First, we can enlarge the symmetry group and consider S^{2}(ℂ^{n})^{∗} ⊗ ℂ^{n} as SL_{n} × SL_{n}-module. Then this module is irreducible and its discriminant variety (the dual variety of the projectivization of the highest weight vector orbit) coincides with D_{1}. Now, consider A = S^{2}V^{∗} ⊗V as SL(V)-module. Then this module is reducible, A = A_{0} +Ã, where A_{0} is a set of algebras with zero trace and Ã is isomorphic to V^{∗} as SL(V)-module. Consider the discriminant of A_{0} as a function on A (forgetting other coordinates). Then the corresponding hypersurface is exactly D_{2}. If we consider the set of linear operators Hom(V, V) = V^{∗} ⊗ V instead of A, then this construction will give us the determinant and the discriminant of the linear operator (see Example 2.18, Theorem 2.5).

Quasiderivations. Let 𝔤 be a Lie algebra with representation ρ : 𝔤 → End(V). Consider any v ∈ V. Then subalgebra

𝔤_{v} = {g ∈ 𝔤 | ρ(g)v = 0}

is called the annihilator of v. The subset

Q𝔤_{v} = {g ∈ 𝔤 | ρ(g)^{2}v = 0}

is called the quasi-annihilator of v. Clearly, 𝔤_{v} ⊂ Q𝔤_{v}. Of course, the quasiannihilator is not a linear subspace in general, However, we have the following version of a Jordan decomposition:

Lemma 10.5 Suppose that ρ is the differential of the representation of an algebraic group. Let g ∈ Q𝔤_{v}. Consider the Jordan decomposition in 𝔤, g = g_{s} +g_{n}, where g_{s} is semisimple and g_{n} is nilpotent, [g_{s}, g_{n}] = 0. Then g_{s} ∈ 𝔤_{v} and g_{n} ∈ Q𝔤_{v}.

The proof is obvious.

For example, suppose that 𝔤 = 𝔤𝔩_{n} and ρ is the natural representation in the vector space V^{∗} ⊗ V^{∗} ⊗ V that parametrizes bilinear multiplications in V. Let A be any algebra. Then the annihilator 𝔤_{A} is identified with the Lie algebra of derivations Der(A). Operators D ∈ Q𝔤_{A} are called quasiderivations. Of course, it is possible to write down explicit equations that determine Q Der(A) = Q𝔤_{A} in End(A), but this formula is quite useless (see [Vi1]). We shall use its particular case that is quite easy to verify:

Lemma 10.6 Let A be an algebra, D ∈ End(A), D^{2} = 0. Then D ∈ Q Der(A) if and only if for any x, y ∈ A we have

D(x)D(y) = D(D(x)y) + D(xD(y)).  (10.1)

Example 10.7 It was conjectured in [Vi1] that all quasiderivations of the algebra of matrices Mat_{n} have the form D(x) = ax+xb, where (a+b)^{2} = [a, b]. Let us give an example of a different quasiderivation. Consider the linear operator D(x) = exe, where e^{2} = 0, e ≠ 0. Then it is to check using (10.1) that D is a quasiderivation. However, D of course can not be written in the form ax + xb.

Quasiderivations can be used to define naive deformations. Namely, suppose that A is any algebra with the multiplication u·v and D is its quasiderivation. Consider the algebra A_{D} with the same underlying vector space and with the multiplication given by u ⋆ v = u · D(v) + D(u) · v − D(u · v). Then if A satisfies any polynomial identity then A_{D} satisfies this identity as well. More generally, let ρ : 𝔤 → End(V) be the differential of a representation of an algebraic group G, v ∈ V, D ∈ Q𝔤_{v}. Suppose that H ⊂ V is a closed conical G-equivariant hypersurface (the vanishing set of a homogeneous Gsemiinvariant) and v ∈ H. Then ρ(D)v also belongs to H. Indeed, since H is equivariant, exp(λρ(D))v belongs to H for any λ ∈ ℂ. Since D is a quasiderivation, exp(λρ(D))v = v + λρ(D)v. Since H is conical, v/λ + ρ(D)v belongs to H. Since H is closed, ρ(D)v also belongs to H.

Now suppose that A is a regular commutative algebra. Then we claim that Q Der(A) = 0.

Theorem 10.8 Let A ∈ A, A ∉ D_{1} ∪ D_{2}. Then Q Der(A) = 0.

Proof. By Lemma 10.5 it is sufficient to check that A has no semisimple derivations and no nilpotent quasiderivations. Suppose that Der(A) contains a non-zero semisimple element. Then the group Aut(A) contains a onedimensional algebraic torus T. Let 𝔱 be its Lie algebra, H ∈ 𝔱, H ≠ 0. We way assume that the spectrum of H in A is integer-valued. Let A_{n} ⊂ A be a weight space of weight n. Then A = ⊕A_{n} is a ℤ-grading. Let v ∈ A be a homogeneous element of a maximal positive (or minimal negative) degree. Then v^{2} = 0, hence A ∈ D_{1}.

Suppose now that E is a non-zero nilpotent quasiderivation. We can embed E in an 𝔰𝔩_{2}-triple ⟨F, H, E⟩ ⊂ End(A). Let

A = ⊕m_{d}R_{d}

be an 𝔰𝔩_{2}-module decomposition, here R_{d} is an irreducible (d+1)-dimensional module. Consider also the weight decomposition

A = ⊕A^{n} and A = ⊕A^{n}

with respect to H. Let

J^{n} = A^{n} + A^{n+1} + A^{n+2} + ….

To avoid the abuse of notations denote by α ∈ A the point corresponding to the algebra A. Let Supp α be the support of α (i.e. all n ∈ ℤ such that α_{n} ≠ 0, where α = ∑ α_{n}, α_{n} ∈ A^{n}). Since E^{2}α = 0, the vector Eα is a linear combination of highest weight vectors, therefore Supp α ⊂ {−1, 0, 1, 2, …}. This is equivalent to J^{n}J^{m} ⊂ J^{n+m−1}. In particular, if v ∈ A is a weight vector of the weight n then v^{2} = 0 as n > 1. Since A is regular, it follows that

A = m_{0}R_{0} ⊕ m_{1}R_{1} = A^{−1} ⊕ A^{0} ⊕ A^{1}.

Since A^{1} = J^{1}, A^{1} is a subalgebra in A. By Theorem 10.4 (2), A^{1} has a one-dimensional subalgebra U spanned by an idempotent u (since A has no quadratic nilpotents). Let v ∈ A^{−1} be a unique vector such that Ev = u. Since E^{2} = 0, we can apply formula (10.1) with x = y = v. We get E(v)^{2} = 2E(E(v)v), hence u = 2E(uv). Therefore, uv − \frac{1}{2}v ∈ J^{0}. Notice that the operator of left multiplication by u preserves J^{0}. Since this operator has an eigenvector in A/J^{0} with eigenvector 1/2, it has such an eigenvector in A. Therefore, U is a singular subalgebra in A. ⊓ ⊔

The following Corollary was first proved in [An] using combinatorial methods.

Corollary 10.9 ([An]) Let A be an n-dimensional semisimple commutative algebra, i.e. A is a direct sum of n copies of ℂ, that is A is the algebra of diagonal n × n matrices. Then A has no nonzero quasiderivations.

Proof. It is sufficient to check that A ∉ D_{1} ∪ D_{2}. Clearly, A has no nilpotents. Suppose that U ⊂ A is a one-dimensional subalgebra spanned by an idempotent e ∈ U. Since the spectrum of the operator of left multiplication by e is integer-valued (actually consists of 0 and 1), it follows that U is not singular. ⊓ ⊔

10.2 Discriminants of Anticommutative Algebras

If the set of certain objects is parametrized by an algebraic variety X (for example, by a vector space) then it makes sense to speak about generic objects. Namely, we say that a generic object satisfies some property if there exists a dense Zariski-open subset X_{0} ⊂ X such that all objects parametrized by points from X_{0} share this property. However, sometimes it is possible to find some discriminant-type closed subvariety Y ⊂ X and then to study properties of ‘regular’ objects parametrized by points from X \ Y. For example, instead of studying generic hypersurfaces it is usually worthy to study smooth hypersurfaces. In this section we implement this program for the study of some quite non-classical object, namely anticommutative algebras, with the multiplication depending on a number of arguments.

Generic anticommutative algebras. Let V = ℂ^{n}. We fix an integer k, 1 < k < n − 1. Let A_{n,k} = Λ^{k}V^{∗} ⊗ V be the vector space of k-linear anticommutative maps from V to V. We identify the points of A_{n,k} with the corresponding algebras, that is, we assume that A ∈ A_{n,k} is the space V equipped with the structure of a k-argument anticommutative algebra. Subalgebras in generic algebras with k = 2 were studied in [T2]. The following theorem is a generalization of these results.

Theorem 10.10 ([T1]) Let A ∈ A_{n,k} be a generic algebra. Then

(i) Every m-dimensional subspace is a subalgebra if m < k. (ii) A contains no m-dimensional subalgebras with k + 1 < m < n. (iii) The set of k-dimensional subalgebras is a smooth irreducible (k−1)(n−k)-

dimensional subvariety in the Grassmanian Gr(k, A). (iv) There are finitely many (k+1)-dimensional subalgebras, and their number

is

∑_{\begin{matrix}n−k−1≥μ_{1}≥...≥μ_{k+1}≥0 \\ n−k−1≥λ_{1}≥...≥λ_{k+1}≥0 \\ μ_{1}≤λ_{1},...,μ_{k+1}≤λ_{k+1} \end{matrix}} (−1)^{|μ|} \frac{(λ_{1} + k)!(λ_{2} + k − 1)! … λ_{k+1}!}{(μ_{1} + k)!(μ_{2} + k − 1)! … μ_{k+1}!}(|λ| − |μ|)!×

× |\frac{1}{(i − j + λ_{j} − μ_{i})!}|_{i,j=1,...,k+1}^{2} ,

where |λ| = λ_{1} + … + λ_{k+1}, |μ| = μ_{1} + … + μ_{k+1}, 1/N! = 0 if N < 0.

(v) A contains a (k + 1)-dimensional subalgebra. (vi) If k = n − 2, then the number of (k + 1)-dimensional subalgebras is equal

to

\frac{2^{n} − (−1)^{n}}{3.}

Proof. The GL_{n}-module A_{n,k} is a sum of two irreducible submodules:

A_{n,k} = A_{n,k}^{0} ⊕ Ã_{n,k}.  (10.2)

Here Ã_{n,k} is isomorphic to Λ^{k−1}V^{∗}: we assign to every (k − 1)-form ω the algebra with multiplication

[v_{1}, …, v_{k}] = ∑_{i=1}^{k}(−1)^{i−1}ω(v_{1}, …, v̂_{i}, …, v_{k})v_{i}.

Note that every subspace of this algebra is a subalgebra. Hence, the lattice of subalgebras of A ∈ A_{n,k} coincides with the lattice of subalgebras of A^{0}, where A ↦ A^{0} is the GL_{n}-equivariant projector on the first summand in (10.2). Algebras in A_{n,k}^{0} will be called zero trace algebras, since A ∈ A_{n,k}^{0} if and only if the (k − 1)-form Tr[v_{1}, …, v_{k−1}, ·] is equal to zero. Hence, the theorem will be proved once we have proved it for generic algebras in A_{n,k}^{0}.

We choose a basis {e_{1}, …, e_{n}} in V, identify GL_{n} with the group of matrices, consider the standard diagonal maximal torus T, and take for B and B_{−} the subgroups of upper- and lower-triangular matrices. We fix an m ≥ k. Consider the parabolic subgroup of matrices

P = ( \begin{matrix}A & 0 \\ ∗ & B \end{matrix} ) ,

where B is an m × m-matrix and A is an (n − m) × (n − m)-matrix. Then G/P coincides with Gr(m, V). Consider the vector bundle

L = Λ^{k}S^{∗} ⊗ V/S

on G/P, where S is the tautological bundle and V/S is the factor-tautological bundle. Then the assumptions of Theorem 4.8 are fulfilled, since L = L_{λ}, where λ is the highest weight of A_{n,k}^{0}. Therefore,

A_{n,k}^{0} = H^{0}(Gr(m, V), Λ^{k}S^{∗} ⊗ V/S).

Let A ∈ A_{n,k}^{0}, and let s_{A} be the corresponding global section. Then the scheme of zeros (Z_{s_{A}})_{red} coincides with the variety of m-dimensional subalgebras of A.

Let us return to the theorem. Statement (i) is obvious. (ii) follows from Theorem 4.8 (i). Theorem 4.8 (ii) implies that if every k-argument anticommutative algebra A contains a k-dimensional subalgebra, then the variety of k-dimensional subalgebras of a generic algebra is a smooth unmixed (k − 1)(n − k)-dimensional subvariety in Gr(k, A). We claim that any (k − 1)- dimensional subspace U can be included in a k-dimensional subalgebra. The multiplication in the algebra defines a linear map from V/U to V/U. Let v + U be a non-zero eigenvector. It is obvious that ℂv ⊕ U is a k-dimensional subalgebra.

We have only to prove that the variety of k-dimensional subalgebras is irreducible. Assume that a section s of the bundle L = Λ^{k}S^{∗} ⊗ V/S over Gr(k, V) corresponding to A has transversal intersection with the zero section. Then the Koszul complex

0 → Λ^{n−k}L^{∗} →^{s} … →^{s} Λ^{2}L^{∗} →^{s} L^{∗} →^{s} O → O_{Z(s)} → 0

is exact by Theorem 3.3.

Note that Λ^{p}L^{∗} is isomorphic to the bundle S^{p}Λ^{k}S ⊗ Λ^{p}(V/S)^{∗}. This is a homogeneous bundle over G/P of the form L_{μ}, where

μ = −ε_{1} − … − ε_{p} + ε_{n−k+1} + … + ε_{n}

and ε_{i} are the weights of the diagonal torus in the tautological representation. Note that the weight μ + ρ (where ρ is the half-sum of the positive roots) is singular (belongs to the wall of the Weil chamber) for any p, 1 ≤ p ≤ n − k. By the Borel–Weil–Bott theorem, H^{∗}(Gr(k, V), Λ^{p}L^{∗}) = 0 for 1 ≤ p ≤ n − k. Hence, H^{0}(Z(s), O_{Z(s)}) = H^{0}(Gr(k, V), O) = ℂ, as was to be shown.

We postpone the proof of assertion (v) till the end of this section and consider assertions (iv) and (vi), that is, we shall calculate the highest Chern class of the bundle Λ^{k}S^{∗} ⊗ V/S over Gr(k + 1, V).

We use the standard notation, facts, and formulae from the Schubert calculus (cf. [Fu1]. The letters λ and μ always denote Young diagrams in the rectangle with k + 1 rows and n − k − 1 columns. It is well known that such diagrams parametrize the basis of the Chow ring of Gr(k + 1, V). The cycle corresponding to λ is denoted by σ_{λ}, σ_{λ} ∈ A^{|λ|}(Gr(k + 1, V)). (We grade the Chow ring by the codimension, |λ| = λ_{1} + … + λ_{k+1}, where λ_{i} is the length of the ith row of λ.) We need the total Chern class of the bundles S and V/S:

c(S) = 1 − σ_{1} + σ_{1,1} − … + (−1)^{k+1}σ_{1,...,1},
c(V/S) = 1 + σ_{1} + σ_{2} + … + σ_{n−k−1}.

We begin by calculating the total Chern class of S ⊗ V/S. The standard formula for the total Chern class of the tensor products of two bundles implies that

c(S ⊗ V/S) = ∑_{μ⊂λ} d_{λμ}Δμ̃(c(S))Δ_{λ′}(c(V/S)),  (10.3)

where

d_{λμ} = |\binom{λ_{i} + k + 1 − i}{μ_{j} + k + 1 − j}|_{1≤i,j≤k+1} , Δ_{λ}(c) = |c_{λ_{i}+j−i}|,

λ′ = (n − k − 1 − λ_{k+1}, n − k − 1 − λ_{k}, …, n − k − 1 − λ_{1}), and μ̃ is the diagram obtained from μ by transposition. We shall use the fact that Δμ̃(c(S)) = Δ_{μ}(s(S)), where s(E) is the Segre class of E. Another fact is that Δ_{λ}(c(V/S)) = σ_{λ}, Δ_{λ}(s(S)) = (−1)^{|λ|}σ_{λ} (since s(S) = 1 − σ_{1} + σ_{2} + … + (−1)^{n−k−1}σ_{n−k−1}). Therefore, (10.3) can be written as

c(S ⊗ V/S) = ∑_{μ⊂λ} d_{λμ}(−1)^{|μ|}σ_{μ}σ_{λ′}.  (10.4)

To calculate the highest Chern class of L = Λ^{k}S^{∗} ⊗ V/S, we use the formula L = Λ^{k+1}S^{∗} ⊗ (S ⊗ V/S). The total Chern class of the first factor is equal to c(Λ^{k+1}S^{∗}) = 1 + σ_{1}. The total Chern class of the second is given by (10.4). Hence, the highest Chern class of L is

c_{top}(L) = ∑_{μ⊂λ} d_{λμ}(−1)^{|μ|}σ_{μ}σ_{λ′}σ_{1}^{|λ|−|μ|}.  (10.5)

The last result of the Schubert calculus that we need is the exact formula for the degree of a product of two cycles. In our case this can be written as

σ_{μ}σ_{λ′}σ_{1}^{|λ|−|μ|} = deg(σ_{μ}σ_{λ′}) = (|λ| − |μ|)! |\frac{1}{(i − j + λ_{j} − μ_{i})!}|_{i,j=1,...,n} ,

(10.6)

where 1/N! = 0 if N < 0. The formula in assertion (iv) of the Theorem can be obtained from (10.5) and (10.6) by a slight modification of the determinant in the formula for d_{λμ}.

It remains to verify the formula in the assertion (vi) of the Theorem. Let k = n − 2. Then the formula in the assertion (iv) can be written as

∑_{0≤i≤j≤k+1}(−1)^{i} \frac{(k + 1)!k! … (k − j + 2)!(k − j)! … 0!}{(k + 1)!k! … (k − i + 2)!(k − i)! … 0!}(j − i)! det^{2}A, (10.7)

where

A = ( \begin{matrix}X & 0 & 0 \\ ∗ & Y & 0 \\ ∗ & ∗ & Z \end{matrix} ) ,

X is an i × i-matrix, Y is a (j − i) × (j − i)-matrix, and the format of Z is a (k + 1 − j) × (k + 1 − j). X and Z are lower-triangular matrices with 1s on the diagonal and Y is given by

Y = ( \begin{matrix}1 & 1 & 0 & 0 & ⋯ & 0 \\ 1/2! & 1 & 1 & 0 & ⋯ & 0 \\ 1/3! & 1/2! & 1 & 1 & ⋯ & 0 \\ 1/4! & 1/3! & 1/2! & 1 & ⋯ & 0 \\ ⋮ & ⋮ & ⋮ & ⋮ & ... & ⋮ \\ 1/(j − i)! & 1/(j − i − 1)! & 1/(j − i − 2)! & 1/(j − 3)! & ⋯ & 1 \end{matrix} ) .

It is easy to verify that det Y = 1/(j − i)!, which enables us to rewrite (10.7) as

∑_{i≤j}(−1)^{i} \frac{(1 + (k − i))! … (1 + (k − j + 1))!}{(k − i)! … (k − j + 1)!}/(j − i)! =

∑_{i≤j}(−1)^{i}\binom{k + 1 − i}{j − i} = ∑_{i}(−1)^{i}2^{k+1−i} = \frac{2^{k+2} − (−1)^{k+2}}{3} = \frac{2^{n} − (−1)^{n}}{3} ,

as was to be shown.

It remains to prove assertion (v) of the Theorem. We have to prove that every A ∈ A_{n,k} has a (k+1)-dimensional subalgebra. We fix a basis {e_{1}, …, e_{n}} in V and consider the subspace U = ⟨e_{n−k}, …, e_{n}⟩. Let M ⊂ A_{n,k} be the subspace that consists of the algebras for which U is a (k + 1)-dimensional subalgebra. We claim that A_{n,k} = GL_{n} ·M. It is sufficient to prove that the differential of the canonical morphism ϕ : GL_{n} ×M → A_{n,k} is surjective at a point (e, A). Consider the algebra A ∈ M in which [e_{n−k}, …, ê_{i}, …, e_{n}] = e_{i} for all n − k ≤ i ≤ n and the other products are zero. We claim that dϕ is surjective at (e, A). Consider the map π : A_{n,k} → A_{n,k}/M. It is sufficient to verify that π ∘ dϕ_{(e,A)}(𝔤𝔩_{n}, 0) = A_{n,k}/M ≃ Λ^{k}U^{∗} ⊗ V/U. But this is obvious, since multiplication in dϕ_{(e,A)}(E_{ji}, 0) for n − k ≤ i ≤ n, 1 ≤ j ≤ n − k − 1 (E_{ji} is the matrix identity) is given by [e_{n−k}, …, ê_{i}, …, e_{n}] = e_{j} with the other products equal to zero. This completes the proof of the theorem. ⊓ ⊔

D-regular algebras. An essential drawback of Theorem 10.10 is the fact that it does not enable us to study the structure of subalgebras of any particular algebra. To correct this, we shall need to introduce some explicit class of ‘regular’ algebras instead of implicit class of generic algebras. The natural way to remove degeneracies is to consider discriminants.

Let A_{n,k}^{0}^{∗} be the GL_{n}-module dual to A_{n,k}^{0}, and let S_{D} be the closure of the orbit of the highest vector, S_{D} ⊂ A_{n,k}^{0}^{∗}. Let P S_{D} ⊂ P A_{n,k}^{0}^{∗} be its projectivization, let P D ⊂ P A_{n,k}^{0} be the subvariety projectively dual to the subvariety P S_{D}, and let D ⊂ A_{n,k}^{0} be the cone over it. We shall call D the D-discriminant subvariety. The algebras A ∈ D are said to be D-singular. The algebras A ∉ D are said to be D-regular.

Theorem 10.11 ([T1])

(i) D is a hypersurface. (ii) Let A be a D-regular algebra. Then the set of k-dimensional subalgebras

of A is a smooth irreducible (k − 1)(n − k)-dimensional subvariety in Gr(k, A).

(iii) Let k = n − 2. Then the degree of D is equal to

\frac{(3n^{2} − 5n)2^{n} − 4n(−1)^{n}}{18.}  (10.8)

Hence, the D-singularity of A is determined by the vanishing of the SL_{n}- invariant polynomial D that defines D. This polynomial is called the Ddiscriminant.

Proof. The fact that D is a hypersurface follows immediately from the results of Section 9.3. Assertion (ii) can be deduced from the corresponding assertion of Theorem 10.10 by an easy calculation with differentials. Namely, let A be a D-regular algebra. We use the arguments in the proof of Theorem 4.8 (ii) and Theorem 10.10 (iii). According to these calculations, it is sufficient to verify that if U is a k-dimensional subalgebra of A, then the map

ψ : 𝔤𝔩_{n} → Λ^{k}U ⊗ A/U,

ψ(g)(v_{1} ∧ … ∧ v_{k}) = g[v_{1}, …, v_{k}] − [gv_{1}, …, v_{k}] − … − [v_{1}, v_{2}, …, gv_{k}] + U

is surjective. Assume the contrary. Then there is a hyperplane H ⊃ U such that the image of ψ lies in Λ^{k}U ⊗ H/U. Consider a non-zero algebra Ã in S_{D} ⊂ A_{n,k}^{0}^{∗} such that [U^{⊥}, V^{∗}, …, V^{∗}] = 0, [V^{∗}, …, V^{∗}] ⊂ H^{⊥}, where U^{⊥} and H^{⊥} are the annihilators of U and H in A_{n,k}^{0}^{∗}. (Such an algebra is unique up to a scalar.) Then Ã annihilates [𝔤𝔩_{n}, A], which is equivalent to the fact that A annihilates [𝔤𝔩_{n}, Ã], that is, the tangent space to S_{D} at Ã. This means that A lies in D, that is, it is a D-singular algebra.

Finally, assertion (iii) follows from Theorem 7.14. ⊓ ⊔

E-regular algebras. We define the E-discriminant and E-regularity only for (n − 2)-argument n-dimensional anticommutative algebras. Let A = A_{n,n−2}^{0}. Consider the projection π : Gr(n − 1, V) × P A → P A on the second summand and the incidence subvariety Z ⊂ Gr(n − 1, V) × P A that consists of pairs S ⊂ P A, where S is a subalgebra in A. Let π̃ = π|_{Z}. By Theorem 10.10, we have π̃(Z) = A. Let Ẽ ⊂ Z be the set of critical points of π̃, let P E = π̃(Ẽ) be the set of critical values of π̃, and let E ⊂ A be the cone over P E. Then E is called the E-discriminant subvariety. The algebras A ∈ E are said to be E-singular. The algebras A ∉ E are said to be E-regular.

Theorem 10.12 ([T1])

(i) E is an irreducible hypersurface. (ii) Let A be an E-regular algebra. Then A has precisely

\frac{2^{n} − (−1)^{n}}{3}

(n − 1)-dimensional subalgebras. (iii) The map π̃ : Ẽ → P E is birational.

Hence, the E-singularity of A is determined by the vanishing of the SL_{n}-invariant polynomial that defines E. This polynomial is called the Ediscriminant. Assertion (iii) can be formulated as follows: a generic E-singular algebra has precisely one “critical” (n − 1)-dimensional subalgebra.

Proof. The proof of assertion (ii) is similar to the proof of Theorem 10.10 (iv). We claim that statement (i) follows from statement (iii). We choose in V^{∗} a basis {f_{1}, …, f_{n}} dual to the basis {e_{1}, …, e_{n}} in V. Let U ∈ Gr(n−1, V) be the hyperplane f_{1} = 0. It is clear that Ẽ = GL_{n} ·M_{0}, where M_{0} = Ẽ∩(U, P A).

Moreover, M = Z ∩ (U, P A) is the linear subspace of the algebras for which U is an (n − 1)-dimensional subalgebra. Let P be the parabolic subgroup of

matrices ( \begin{matrix}A & 0 \\ ∗ & B \end{matrix} ), and let 𝔲 be the Lie algebra of matrices ( \begin{matrix}0 & X \\ 0 & 0 \end{matrix} ), where

B is an (n−1)×(n−1)-matrix and X is an 1×(n−1)-matrix. Every algebra A ∈ M defines a linear map 𝔲 → Λ^{n−1}U^{∗} ⊗ V/U. Since A ∈ M_{0} if and only if this map is degenerate, we have codim_{M} M_{0} = 1. It is obvious that M_{0} is irreducible, since M_{0} is the spreading of the subspace

M_{0}^{1} = {A ∈ M_{0} | the algebra E_{12}A has a subalgebra U}

by the group P. Hence, Ẽ is an irreducible divisor in Z, and assertion (i) follows from assertion (iii).

We prove (iii). We shall say that an (n − 1)-dimensional subalgebra U′ of A ∈ E is critical if (U′, A) lies in Ẽ. In this case there is an (n−2)-dimensional subspace W′ ⊂ U′ such that if V = U′ ⊕ ℂe and v ∈ 𝔤𝔩_{n} is a non-zero linear operator such that v(V) ⊂ ℂe and v(W′) = 0, then U′ is a subalgebra of vA. To prove assertion (iii) it is sufficient to prove that in generic algebras in M_{0}^{1} the subalgebra U is the unique critical subalgebra. Let N ⊂ M_{0}^{1} be the subvariety of all algebras that have another critical subalgebra.

Note that M_{0}^{1} is normalized by the parabolic subgroup

Q = ( \begin{matrix}∗ & 0 & 0 & … & 0 \\ ∗ & ∗ & 0 & … & 0 \\ ∗ & ∗ & ∗ & … & ∗ \\ ⋮ & ⋮ & ⋮ & ... & ⋮ \\ ∗ & ∗ & ∗ & … & ∗ \end{matrix} ) .

Then N is the spreading of the subvarieties N_{2} and N_{3} by the group Q, where A ∈ N_{i} if and only if the hyperplane f_{i} = 0 is a critical subalgebra. In turn, N_{2} is the spreading of the vector spaces N_{2}^{1} and N_{2}^{3}, and N_{3} is the spreading of the subspaces N_{3}^{1}, N_{3}^{2}, and N_{3}^{4}, where N_{i}^{j} ⊂ N_{i} is the subspace of all algebras such that the (n − 2)-dimensional subspace W′ (mentioned above) is given by f_{i} = f_{j} = 0. Let Q_{i}^{j} ⊂ Q be the subgroup that normalizes the flag f_{i} ⊂ ⟨f_{i}, f_{j}⟩. It is easy to verify that

codim_{Q} Q_{2}^{1} = 1, codim_{Q} Q_{2}^{3} = n − 1, codim_{Q} Q_{3}^{1} = n − 1,
codim_{Q} Q_{3}^{2} = n, codim_{Q} Q_{3}^{4} = 2n − 3.

On the other hand,

codim_{M_{0}^{1}} N_{2}^{1} = n,

codim_{M_{0}^{1}} N_{2}^{3} = codim_{M_{0}^{1}} N_{3}^{1} = codim_{M_{0}^{1}} N_{3}^{2} = codim_{M_{0}^{1}} N_{3}^{4} = 2n − 2.

Hence, codim_{M_{0}^{1}} QN_{i}^{j} ≥ 1 in all cases, which completes the proof of the theorem. ⊓ ⊔ Regular 4-dimensional anticommutative algebras. An (n − 2)-argu-ment n-dimensional anticommutative algebra is said to be regular if it is D-regular and E-regular. In this subsection we consider 2-argument 4dimensional algebras. The corresponding generic algebras were studied in [T2].

Theorem 10.13 ([T1]) Let A be a 4-dimensional regular anticommutative algebra. Then

(i) A has precisely five 3-dimensional subalgebras. The set of these subalgebras is a generic configuration of five hyperplanes. In particular, A has a pentahedral normal form, that is, it can be reduced by a transformation that belongs to GL_{4} to an algebra such that the set of its five subalgebras is a Sylvester pentahedron x_{1} = 0, x_{2} = 0, x_{3} = 0, x_{4} = 0, x_{1} + x_{2} + x_{3} + x_{4} = 0.

(ii) A has neither one- nor two-dimensional ideals. (iii) The set of two-dimensional subalgebras of A is a del Pezzo surface of

degree 5 (a blowing up of ℙ^{2} at four generic points). (iv) A has precisely 10 fans, that is, flags V_{1} ⊂ V_{3} of 1-dimensional and 3-

dimensional subspaces such that every intermediate subspace U, V_{1} ⊂ U ⊂ V_{3}, is a two-dimensional subalgebra.

Proof. We begin with assertion (i). We have to prove that if A is a 4dimensional regular anticommutative algebra with zero trace, then the set of its three-dimensional subalgebras S_{1}, S_{2}, …, S_{5} is a generic configuration of hyperplanes, that is, the intersection of any three of them is one-dimensional and the intersection of any four of them is zero-dimensional. Indeed, assume, for example, that U = S_{1} ∩ S_{2} ∩ S_{3} is two-dimensional. Let v ∈ U and v ≠ 0. Then [v, ·] induces a linear operator on A/U, since U is a subalgebra. S_{1}/U, S_{2}/U, and S_{3}/U are one-dimensional eigenspaces. Since dim A/U = 2, the operator is a dilation. Since this is true for any v ∈ U, any three-dimensional subspace that contains U is a three-dimensional subalgebra, which contradicts the fact that there are precisely five such subalgebras.

Now assume that U = S_{1} ∩ S_{2} ∩ S_{3} ∩ S_{4} is one-dimensional, and let v ∈ U, v ≠ 0. Then the operator [v, ·] induces an operator on A/U. This operator has four two-dimensional eigenspaces S_{1}/U, …, S_{4}/U of which any three have zero intersection. Hence, this operator is a dilation. Let W ⊃ U be an arbitrary two-dimensional subspace, and let w ∈ W be a vector that is not proportional to v. Then the operator [w, ·] induces a linear operator on A/W. Let z be a non-zero eigenvector. Then ⟨v, w, z⟩ is a three-dimensional subalgebra. Therefore, every vector can be included in a three-dimensional subalgebra, which contradicts the fact that there are only five such subalgebras.

This argument also shows that A has no one-dimensional ideals. Since every three-dimensional subspace that contains a two-dimensional ideal is a subalgebra, there are no two-dimensional ideals, which completes the proof of assertion (ii).

To prove assertion (iii), we consider the subvariety X of two-dimensional subalgebras in A. Then X ⊂ Gr(2, 4). Consider the Plücker embedding Gr(2, 4) ⊂ ℙ^{5} = P (Λ^{2}ℂ^{4}). First we claim that the embedding X ⊂ ℙ^{5} is nondegenerate, that is, the image is contained in no hyperplane. Let S_{1}, S_{2}, S_{3}, S_{4} be four three-dimensional subalgebras. Since it is a generic configuration, we can choose a basis {e_{1}, e_{2}, e_{3}, e_{4}} in A such that S_{i} = ⟨e_{1}, …, ê_{i}, …, e_{4}⟩. Since the intersection of three-dimensional subalgebras is a two-dimensional subalgebra, A has six subalgebras ⟨e_{i}, e_{j}⟩, i ≠ j. The set of corresponding bivectors e_{i} ∧ e_{j} is a basis in Λ^{2}ℂ^{4}. Therefore, they can lie in no hyperplane. Simple calculation with Koszul complexes (cf. Theorem 3.3) shows that H^{0}(X, O_{X}(1))^{∗} = Λ^{2}ℂ^{4}.

To prove that X is a del Pezzo surface of degree five, we have only to verify that O_{X}(1) coincides with the anticanonical sheaf (see [Ma]). Since Y = Gr(2, 4) is a quadric in ℙ^{5}, we have

ω_{Y} = O_{Y} (2 − 5 − 1) = O_{Y} (−4).

The set X is a non-singular subvariety of codimension 2 in Y. Therefore, ω_{X} = ω_{Y} ⊗ Λ^{2}N_{X/Y}, where N_{X/Y} is the normal sheaf. Further, X is the scheme of zeros of a regular section of the vector bundle L = Λ^{2}S^{∗} ⊗ V/S, whence N_{X/Y} = L|_{Y} and Λ^{2}N_{X/Y} = O_{X}(3), since c_{1}(L) = 3H. We obtain that ω_{X} = O_{X}(−4) ⊗ O_{X}(3) = O_{X}(−1), as was to be shown.

It remains to prove assertion (iv). Since X is a del Pezzo surface of degree five, it contains ten straight lines. Since the embedding X ⊂ P (Λ^{2}ℂ^{4}) is anticanonical, these straight lines are ordinary straight lines in P (Λ^{2}ℂ^{4}) that lie in Gr(2, 4). It remains to establish a bijection between these straight lines and fans. If b ∈ Λ^{2}ℂ^{4}, then b belongs to the cone over Gr(2, 4) if and only if b ∧ b = 0. If b_{1} and b_{2} belong to this cone, then the straight line that joins them belongs to this cone if and only if b_{1} ∧ b_{2} = 0, which coincides with the fan condition. ⊓ ⊔

Dodecahedral section. Let us start with some definitions. Let X be an irreducible G-variety (a variety with an action of algebraic group G), S ⊂ X be an irreducible subvariety. Then S is called a section of X if \overline{G · S} = X. The section S is called a relative section if the following condition holds: there exists a dense Zariski-open subset U ⊂ S such that if x ∈ U and gx ∈ S then g ∈ H, where H = N_{G}(S) = {g ∈ G | gS ⊂ S} is the normalizer of S in G (see [PV]). In this case for any invariant function f ∈ ℂ(X)^{G} the restriction f|_{S} is well-defined and the map

ℂ(X)^{G} → ℂ(S)^{H}, f ↦ f|_{S},

is an isomorphism. Any relative section defines a G-equivariant rational map ψ : X → G/H: if g^{−1}x ∈ S then x ↦ gH. Conversely, any G-equivariant rational map ψ : X → G/H with irreducible fibers defines the relative section \overline{ψ^{−1}(eH)}.

We are going to apply Theorem 10.13 and to construct a relative section in the SL_{4}-module A_{0} (the module of 4-dimensional anticommutative algebras with zero trace). The action of SL_{4} on ‘Sylvester pentahedrons’ is transitive with finite stabilizer H (which is the central extension of the permutation group S_{5}). In the sequel the Sylvester pentahedron will always mean the standard configuration formed by the hyperplanes

x_{1} = 0, x_{2} = 0, x_{3} = 0, x_{4} = 0, x_{1} + x_{2} + x_{3} + x_{4} = 0.

Let S ⊂ A_{0} be a linear subspace formed by all algebras such that the hyperplanes of Sylvester pentahedron are their subalgebras. Then Theorem 10.13 implies that S is a 5-dimensional linear relative section of SL_{4}-module A_{0}.

It is easy to see that the multiplication in algebras from S is given by formulas

[e_{i} , e_{j}] = a_{ij}e_{i} + b_{ij}e_{j} (1 ≤ i < j ≤ 4),

where a_{ij} and b_{ij} satisfy the certain set of linear conditions. Consider 6 algebras A_{1}, …, A_{6} with the following structure constants

A_{1}  A_{2}  A_{3}  A_{4}  A_{5}  A_{6}
a_{12}  0  1  −1  1  0  −1
b_{12}  1  0  1  −1  −1  0
a_{13}  1  1  −1  0  −1  0
b_{13}  0  −1  0  1  1  −1
a_{14}  1  0  0  1  −1  −1
b_{14}  −1  1  −1  0  0  1
a_{23}  −1  −1  0  1  0  1
b_{23}  1  0  −1  0  1  −1
a_{24}  0  −1  −1  0  1  1
b_{24}  −1  1  0  1  −1  0
a_{34}  −1  1  1  −1  0  0
b_{34}  0  0  −1  1  −1  1

Then it is easy to see that A_{i} ∈ S for any i. Moreover, algebras A_{i} satisfy the unique linear relation A_{1} + … + A_{6} = 0. It follows that any A ∈ S can be written uniquely in the form α_{1}A_{1} + … + α_{6}A_{6}, where α_{1} + … + α_{6} = 0. The coordinates α_{i} are called dodecahedral coordinates and S is called the dodecahedral section (this name will be clear later).

The stabilizer of the standard Sylvester pentahedron in PGL_{4} is isomorphic to 𝕊_{5} represented by permutations of its hyperplanes. The group S_{5} is generated by the transposition (12) and the cycle (12345). The preimages of these elements in GL_{4} are given by matrices

σ = ( \begin{matrix}0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{matrix} ) τ = ( \begin{matrix}−1 & −1 & −1 & −1 \\ 1 & 0 & 0 & 0 \\ 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \end{matrix} ) .

The preimage of S_{5} in SL_{4} is the group H of 480 elements. The representation of H in S induces the projective representation of S_{5} in ℙ^{4}. We have the following

Proposition 10.14 This projective representation is the projectivization of the 5-dimensional irreducible representation of S_{5} (any of two possible).

Proof. Recall certain ‘folklore’ facts about the representation theory of S_{5}. It is well-known that S_{5} admits exactly two embeddings in S_{6} up to conjugacy. One is standard via permutations of the first five elements of the six-element set permuted by S_{6}. The second one can be obtained from the first one by taking the composition with the unique (up to conjugacy) outer involution of S_{6}. S_{5} has exactly two irreducible 5-dimensional representations, which have the same projectivizations. One of these representations has the following model. One takes the tautological 5-dimensional irreducible representation of S_{6} and consider its composition with the non-standard embedding S_{5} ⊂ S_{6}.

Now let us return to our projective representation. The action of σ and τ on algebras A_{i} is given by formulas

( \begin{matrix}σA_{1} \\ σA_{2} \\ σA_{3} \\ σA_{4} \\ σA_{5} \\ σA_{6} \end{matrix} ) = ( \begin{matrix}−A_{2} \\ −A_{1} \\ −A_{4} \\ −A_{3} \\ −A_{6} \\ −A_{5} \end{matrix} )  and  ( \begin{matrix}τA_{1} \\ τA_{2} \\ τA_{3} \\ τA_{4} \\ τA_{5} \\ τA_{6} \end{matrix} ) = ( \begin{matrix}A_{1} \\ A_{6} \\ A_{2} \\ A_{3} \\ A_{4} \\ A_{5} \end{matrix} ) .  (10.9)

Therefore, S_{5} permutes the lines spanned by A_{i}. Moreover, the induced embedding is clearly non-standard: transposition in S_{5} maps to the composition of 3 independent transpositions in S_{6}. It is clear that that the corresponding projective representation of S_{5} is isomorphic to the projectivization of the 5-dimensional irreducible representation in the model described above.

⊓ ⊔

Remark 10.15 The section S is called dodecahedral by the following reason. Though the surjection H → S_{5} does not split, the alternating group A_{5} can be embedded in H. The induced representation of A_{5} in S has the following description. 𝔸_{5} can be realised as a group of rotations of dodecahedron. Let {Γ_{1}, …, Γ_{6}} be the set of pairs of opposite faces of dodecahedron. Consider the vector space of functions

f : {Γ_{1}, …, Γ_{6}} → ℂ, ∑_{i=1}^{6} f(Γ_{i}) = 0.

Then this vector space is an A_{5}-module. It is easy to see that this module is isomorphic to S via the identification A_{i} ↦ f_{i}, where f_{i}(Γ_{i}) = 5, f_{i}(Γ_{j}) = −1, j ≠ i.

The following proposition follows from the discussion above

Theorem 10.16 The restriction of invariants induces an isomorphism of invariant fields

ℂ(A_{0})^{GL_{4}} ≃ ℂ(ℂ^{5})^{ℂ^{∗}×𝕊_{5}},

where ℂ^{∗} acts on ℂ^{5} by homotheties and 𝕊_{5} acts via any of two 5-dimensional irreducible representations.

Remark 10.17 The Sylvester pentahedron also naturally arises in the theory of cubic surfaces. The SL_{4}-module of cubic forms S^{3}(ℂ^{4})^{∗} admits the relative section (the so-called Sylvester section, or Sylvester normal form). Namely, a generic cubic form in a suitable system of homogeneous coordinates x_{1}, …, x_{5}, x_{1} + … + x_{5} = 0, can be written as a sum of 5 cubes x_{1}^{3} + … + x_{5}^{3}. The Sylvester pentahedron can be recovered from a generic cubic form f in a very interesting way: its 10 vertices coincide with 10 singular points of a quartic surface det Hes(f). The Sylvester section has the same normalizer H as our dodecahedral section. It can be proved [Bek] that in this case the restriction of invariants induces an isomorphism

ℂ(S^{3}(ℂ^{4})^{∗})^{GL_{4}} ≃ ℂ(ℂ^{5})^{ℂ^{∗}×𝕊_{5}},

where ℂ^{∗} acts via homotheties and 𝕊_{5} via permutations of coordinates (i.e. via the reducible 5-dimensional representation). Other applications of the Sylvester pentahedron to moduli varieties can be found in [Bar].

These results were used in [T2] in order to prove that the field of invariant functions of the 5-dimensional irreducible representation of S_{5} is rational (is isomorphic to the field of invariant functions of a vector space). From this result it is easy to deduce that in fact the field of invariant functions of any representation of S_{5} is rational (see [T5]).

10.3 Self–dual Varieties

10.3.A Self–dual Polarized Flag Varieties

A projective variety X ⊂ ℙ^{n} is called self-dual if X is isomorphic to X^{∗} as an embedded projective variety, i.e. there exists an isomorphism f : ℙ^{N} → ℙ^{n}^{∗} such that f(X) = X^{∗}. A lot of examples is provided by the Pyasetskii pairing 2.9. Indeed, if G is a connected reductive algebraic group acting on a vector space V then the dual action G : V^{∗} can be described as follows.

Having passed to a finite covering of G, one may assume that there is an involution θ ∈ Aut(G) (a so-called involution of maximal rank) such that θ(t) = t^{−1} for any t ∈ T, where T is the fixed maximal torus in G. Along with the action of G on V we may consider the twisted action given by g ⋆ v = ω(g)v. Then it is easy to see that the twisted action is isomorphic to the dual action G : V^{∗}. Therefore, we may canonically identify G-orbits in V and in V^{∗}. In particular, if G acts on V with finitely many orbits then we may consider the Pyasetskii pairing (hence the projective duality of projectivizations) as pairing between G-orbits in V.

For example, let L be a simple algebraic group and P a parabolic subgroup with Abelian unipotent radical. In this case 𝔩 = Lie L admits a short ℤgrading with only three non-zero parts:

𝔩 = 𝔩_{−1} ⊕ 𝔩_{0} ⊕ 𝔩_{1}.

Here 𝔩_{0}⊕𝔩_{1} = Lie P and exp(𝔩_{1}) is the Abelian unipotent radical of P. Let G ⊂ L be a reductive subgroup with Lie algebra 𝔩_{0}. Recall that by Theorem 2.17 G has finitely many orbits in 𝔩_{1} naturally labelled by integers from the segment [0, r] such that O_{k} corresponds to O_{r−k} via Pyasetskii pairing (here we use the above identifications of G-orbits in 𝔩_{1} and 𝔩_{−1}). Therefore, if r is even then the closure of the projectivization of O\frac{r}{2} is a self-dual projective variety.

However, it is well-known that the closure of the projectivization of O_{i} is smooth if and only if i = 1 or i = r (in the last case \overline{ℙ(O_{r})} = ℙ(𝔩_{1})), see e.g. [Pan] for the equivariant resolution of singularities of \overline{O_{i}}. Therefore, the construction above gives a smooth self-dual projective variety if and only if r = 2. It is worthy to write down all arising cases. We use the notation from Section 2.3.

Example 10.18 Consider the short gradings of 𝔩 = 𝔰𝔩_{n+2}. Then O_{1} is the variety of n × 2-matrices of rank 1. The projectivization of O_{1} is identified with X = ℙ^{n−1} × ℙ^{1} in the Segre embedding.

Example 10.19 Consider the short grading of 𝔩 = 𝔰𝔬_{n+2} that corresponds to β being the first simple root. The projectivization of O_{1} is identified with the quadric hypersurface in ℙ^{n−1}. The quadric hypersurface in ℙ^{5} also arises from the short grading of 𝔩 = 𝔰𝔬_{8} = D_{4} that corresponds to β = α_{1}, β = α_{3}, or β = α_{4} (Plücker quadric).

Example 10.20 Consider the short grading of 𝔩 = 𝔰𝔬_{10} = D_{5} that corresponds to β = α_{4} or β = α_{5}. The projectivization of O_{1} is identified with X = Gr(2, 5) in the Plücker embedding.

Example 10.21 The short grading of E_{6} gives the spinor variety 𝕊_{5}.

Remarkably, there are no other known examples of smooth non-linear self-dual varieties. Moreover, it is widely expected that this list is complete. At least, this is true for polarized flag varieties:

Theorem 10.22 ([Sn]) Let X = G/P ⊂ ℙ^{N} be a non-linear polarized flag variety. If dim X = dim X^{∗} the X is one of the following

– A quadric hypersurface.

– The Segre embedding of ℙ^{n} × ℙ^{1}.

– The Plücker embedding of Gr(2, 5).

– The 10-dimensional spinor variety 𝕊_{5}.

Proof. If dim X = dim X^{∗} then either X is a hypersurface or def X > 0. In the first case X is necessarily a quadric hypersurface. Indeed, let Y ⊂ ℙ^{N}^{∗} be the projectivization of the highest weight vector orbit of the dual representation. Then Y is isomorphic to X as an embedded projective variety. The intersection Y^{∗} ∩ X is non-empty and G-invariant, therefore, Y^{∗} = X and X is a self-dual smooth hypersurface. Then X is a quadric hypersurface by Example 7.4. An alternative proof follows from the fact that the closure of the highest weight vector orbit is always cut out by quadrics [Li].

If def X > 0 then the claim easily follows from Theorem 9.21. ⊓ ⊔ Theorem 10.22 was used in [Sn] to recover the classification of homoge-

neous real hypersurface in a complex projective space due to [Ta].

Theorem 10.23 ([Ta, Sn]) Let M be a homogeneous complete real hypersurface embedded equivariantly in ℙ^{N}. Then M is a tube over a linear projective space or one of the 4 self-dual homogeneous spaces X ⊂ ℙ^{N} listed in Theorem 10.22.

Sketch of the proof. Let M = K/L, where K is a compact Lie group. The main idea is to use the fact that M is necessarily a tube over a complex submanifold X ⊂ ℙ^{N} called a focal submanifold, see e.g. [CR]. Then X is easily seen to be homogeneous, therefore X is a flag variety, X = G/P, where G is the complexification of K. Moreover, G acts transitively on the normal directions to X. It follows that the conormal variety ℙ(N_{X}ℙ^{N}) is also homogeneous, therefore the dual variety X^{∗} is also homogeneous being the image of the conormal variety. Hence X^{∗} is smooth and therefore dim X = dim X^{∗} by Theorem 5.13. Now we can apply Theorem 10.22. ⊓ ⊔

10.3.B Around Hartshorne Conjecture

In [Ha2] R. Hartshorne has suggested a number of conjectures related to the geometry of projective varieties of small codimension. This work has stimulated a serie of remarkable researches, for example the Zak’s proofs of Theorem 5.17 on linear normality and Theorem 5.19 on Severi varieties. Undoubtfully, the most famous conjecture from this paper is the so-called Hartshorne conjecture on complete intersections:

Conjecture 10.24 ([Ha2]) If X is a smooth n-dimensional projective variety in ℙ^{N} and codim X < N/3, then X is a complete intersection.

This conjecture is still very far from being solved. Only partial results are known, for example the following Landsberg’s theorem on complete intersections. The proof is based on the technique of Section 6.4.

Theorem 10.25 ([Lan1]) Let X ⊂ ℙ(V) = ℙ^{N} be an irreducible projective variety cut out by quadrics, that is, the homogeneous ideal I_{Cone(X)} in ℂ[V] is generated by its second homogeneous component I_{Cone(X)}^{2}. Let b = dim Sing X

(set b = −1 if X is smooth). If codim X < \frac{N − (b + 1) + 3}{4}, then X is a

complete intersection.

If Hartshorne conjecture is true then any smooth projective variety X in ℙ^{N} such that codim X < N/3 is a complete intersection. Therefore, X^{∗} should be a hypersurface by Theorem 4.2. It is not known whether it is true or not. In particular, if X is a smooth self-dual variety then, up to Hartshorne Conjecture, either X is a quadric hypersurface or codim X ≥ N/3. In particular, the following theorem should give a complete list of smooth self-dual varieties:

Theorem 10.26 ([E1]) Let X be a nonlinear smooth projective variety in ℙ^{N}. Assume that codim X ≥ N/3. Suppose that dim X = dim X^{∗}. Then X is one of the following varieties:

(i) X is a hypersurface in ℙ^{2} or ℙ^{3}. (ii) X is the Segre embedding of ℙ^{1} × ℙ^{n−1} in ℙ^{2n−1}. (iii) X is the Plücker embedding of Gr(2, 5) in ℙ^{9}. (iv) X is the 10-dimensional spinor variety 𝕊_{5} in ℙ^{15}.

Sketch of the proof. Let dim X = n. We may assume that n ≥ 3 by Example 1.13 and Example 6.16. Now def X = N − 1 − n. Since def X ≤

n − 2 by Example 6.17, we conclude that n ≥ \frac{N + 1}{2}. If n = \frac{N + 1}{2}, then

def X = n − 2 and, therefore, we have case (ii) by Example 6.18. So we

may assume that n ≥ N/2 + 1. Then K_{X} = O_{X}(\frac{−N − 1}{2}) by Theorem 6.19

(d). We conclude that def X = N − 1 − n ≤ \frac{n − 2}{2} by Theorem 6.23 (a).

Hence n ≥ 2N/3. By our assumption n ≤ 2N/3. Therefore, n = 2N/3. Now def X = N − 1 − n = \frac{1}{2}n − 1. Therefore n is even. Since def X ≡ n mod 2 by Theorem 6.15, we conclude that n = 4m + 2 and def X = 2m. It follows that m ≤ 2 by Theorem 6.23 (b). This leaves two cases: X is either a 6dimensional variety in ℙ^{9} or a 10-dimensional variety in ℙ^{15}. A very delicate treatment of both cases can be found in [E1]. ⊓ ⊔ 10.3.C Self–dual Nilpotent Orbits

If X ⊂ ℙ^{n} is a smooth projective variety then X is almost never self-dual. Moreover, up to Hartshorne conjecture Theorem 10.26 provides a complete list of them. However, there are a lot of non-smooth self-dual varieties. Many equivariant self-dual varieties are provided by Pyasetskii Theorem 2.9. Perhaps, the most interesting examples of self-dual varieties are the Kummer surface in ℙ^{3} [GH1] and the Coble quartic in ℙ^{7} [Pau].

Another interesting examples were found in [Pop]. Let G be a connected reductive group acting linearly on a vector space V. Suppose that there exists a G-invariant non-degenerate scalar product (·, ·) on V, in particular we have a natural isomorphism V ≃ V^{∗}. Finally, suppose that G acts on the null-cone

R(V) = \{v ∈ V | \overline{G · v} ∋ 0\}

with finitely many orbits. In particular, any G-orbit O ⊂ R(V) is conical.

Theorem 10.27 ([Pop]) Let O = G · v ⊂ R(V) be a non-zero orbit, X = \overline{ℙ(O)} be the closure of its projectivization. Then X = X^{∗} if and only if (𝔤 · v)^{⊥} ⊂ R(V).

Proof. Since O is conical, v ∈ 𝔤 · v. Therefore, 𝔤 · v is the affine cone Cone(T̂_{[v]}X) over the embedded tangent space T̂_{[v]}X. Hence, X^{∗} is equal to \overline{ℙ(G · (𝔤 · v)^{⊥})}. Since (·, ·) is G-invariant, we have v ∈ (𝔤 · v)^{⊥}, therefore, X ⊂ X^{∗}. If X = X^{∗} then, obviously, (𝔤 · v)^{⊥} ⊂ R(V). Suppose that (𝔤 · v)^{⊥} ⊂ R(V). Then X^{∗} ⊂ ℙ(R(V)). Therefore, X^{∗} = \overline{ℙ(O′)} for some orbit O′ ⊂ R(V). Applying the same arguments we get that X^{∗} ⊂ X^{∗}^{∗}. By Reflexivity Theorem, we finally obtain X = X^{∗}. ⊓ ⊔

Let G be semisimple, V = 𝔤 be the adjoint representation. Then the Killing form (·, ·) is G-invariant, R(𝔤) is the cone of nilpotent elements. Since there are finitely many nilpotent orbits, we can apply the previous Theorem. For any x ∈ 𝔤 the orthogonal complement (ad(𝔤)x)^{⊥} is identified with the centralizer 𝔤_{x} of x in 𝔤. Therefore, a nilpotent orbit O = Ad(G)x has a selfdual projectivization if and only if the centralizer 𝔤_{x} belongs to R(𝔤), i.e. has no semi-simple elements. These orbits are called the distinguished nilpotent orbits. For example, the regular nilpotent orbit (the orbit dense in R(𝔤)) is distinguished, hence the null-cone R(𝔤) itself has a self-dual projectivization.

The distinguished nilpotent orbits are, in a sense, the building blocks for the set of all nilpotent orbits. Namely, due to the Bala–Carter correspondence [BCa1, BCa2] the set of all nilpotent orbits in a semisimple Lie algebra 𝔤 is in the natural bijection with the set of isoclasses of pairs (𝔩, O), where 𝔩 ⊂ 𝔤 is a Levi subalgebra and O ⊂ 𝔩 is a distinguished nilpotent orbit in [𝔩, 𝔩].

10.4 Linear Systems of Quadrics of Constant Rank

Theorem 10.28 ([IL]) Let X ⊂ ℙ^{N} be a smooth n-dimensional variety. If H ∈ X^{∗}_{sm} then the projective second fundamental form |II|_{H} of X^{∗} at H is a system of quadrics of projective dimension def X and constant rank n − def X.

This Theorem (analogous to Theorem 6.12) leads to the examination of linear systems of quadrics of constant rank.

More generally, let V = ℂ^{m}, W = ℂ^{n} and let A ⊂ Hom(V, W) (resp. A ⊂ S^{2}V^{∗}, resp. A ⊂ Λ^{2}V^{∗}) be a linear subspace.

Definition 10.29 A is said to be of bounded rank r if for all f ∈ A, rank f ≤ r. A is said to be of constant rank r if for all non-zero f ∈ A, rank f = r. A is said to be of rank bounded below by r if for all non-zero f ∈ A, rank f ≥ r.

We define the numbers

l(r, m, n) = max{dim A | A ⊂ Hom(V, W) is of constant rank r}

\underline{l}(r, m, n) = max\{dim A | A ⊂ Hom(V, W) is of bounded rank r\}

\overline{l}(r, m, n) = max\{dim A | A ⊂ Hom(V, W) is of rank bounded below by r\}

Similarly, define c(r, m), \underline{c}(r, m), and \overline{c}(r, m) in the symmetric case and λ(r, m), \underline{λ}(r, m), and \overline{λ}(r, m) in the skew-symmetric case. Recall that any skew-symmetric matrix has even rank.

Proposition 10.30 ([IL]) The upper bounds for dimensions of linear subspaces of matrices of rank bounded below by r are as follows.

(a) \overline{l}(r, m, n) = (m − r)(n − r);

(b) \overline{c}(r, m) = \binom{m−r+1}{2};

(c) \overline{λ}(r, m) = \binom{m−r}{2}, r even.

Proof. Let

X_{r} = {f ∈ ℙ(Hom(V, W)) | rank f ≤ r}

and similarly for X_{r}^{c} ⊂ ℙ(S^{2}V) and X_{r}^{λ} ⊂ ℙ(Λ^{2}V). Clearly, \overline{l}(r, m, n) = codim X_{r}, \overline{c}(r, m) = codim X_{r}^{c}, and \overline{λ}(r, m) = codim X_{r}^{λ}. Now the theorem follows by an easy dimension count. ⊓ ⊔

Given a linear subspace A ⊂ Hom(V, W) we can also consider A as a linear subspace A^{c} of S^{2}(V^{∗} ⊕ W) or a linear subspace A^{λ} of Λ^{2}(V^{∗} ⊕ W). In matrix form, for any matrix a ∈ A we associate a symmetric matrix a^{c} ∈ A^{c} or skew-symmetric matrix a^{λ} ∈ A^{λ} of the form

a^{c} = ( \begin{matrix}0 & a \\ a^{t} & 0 \end{matrix} ) , a^{λ} = ( \begin{matrix}0 & a \\ −a^{t} & 0 \end{matrix} ) .

If A has constant rank r (resp. bounded rank r, resp. rank bounded below by r) then A^{c} and A^{λ} have constant rank 2r (resp. bounded rank 2r, resp. rank bounded below by 2r). A^{c} and A^{λ} are called doublings of A.

Proposition 10.31 ([IL]) The lower bounds for dimensions of linear subspaces of matrices of constant rank are as follows.

(a) If 0 < r ≤ m ≤ n then l(r, m, n) ≥ n − r + 1.

(b) If r ≥ 2 is even then c(r, m) ≥ m − r + 1 and λ(r, m) ≥ m − r + 1.

Proof. X_{r−1} ⊂ ℙ(Hom(ℂ^{r}, ℂ^{n}) has codimension n − r + 1. Thus we can find an (n−r+1)-dimensional subspace A ⊂ Hom(ℂ^{r}, ℂ^{n}) of constant rank r and (a) follows. The doublings of A produce (n − r + 1)-dimensional subspaces A^{c} ⊂ S^{2}ℂ^{r+n} and A^{λ} ⊂ Λ^{2}ℂ^{r+n} of constant rank 2r. This implies (b). ⊓ ⊔

Any linear map ψ : A → Hom(V, W) can be viewed as an element of

A^{∗} ⊗ Hom(V, W) = H^{0}(ℙ(A), O_{ℙ(A)}(1)) ⊗ Hom(V, W)
= H^{0}(ℙ(A), O_{ℙ(A)}(1) ⊗ Hom(V, W)).

Thus to give a linear map A → Hom(V, W) (say, embedding) is the same as to give a vector bundle map

\overline{ψ} : V ⊗ O_{ℙ(A)} → W ⊗ O_{ℙ(A)}(1)

on ℙ(A). Let K, N, and E be the kernel, the cokernel, and the image of \overline{ψ}. Then it is easy to see that ψ is an embedding of constant rank r if and only if all K, N, and E are vector bundles of ranks dim V − r, dim W − r, and r if and only if any of K, N, and E is a vector bundle of rank as above. Therefore, all embeddings A ⊂ Hom(V, W) of constant rank are in one-one correspondence with exact sequences of vector bundles

0 → K → O_{ℙ(A)} ⊗ V → E → 0 and 0 → E → O_{ℙ(A)}(1) ⊗ W → N → 0.

Similarly, all embeddings A ⊂ S^{2}V (resp. A ⊂ Λ^{2}V) of constant rank are in one-one correspondence with exact sequences of vector bundles

0 → K → O_{ℙ(A)} ⊗ V^{∗} → E → 0,

where E is a rank r vector bundle such that E ≃ E^{∗}(1) and this isomorphism is symmetric (resp. skew-symmetric).

Theorem 10.32 ([IL])

(a) Let A ⊂ S^{2}V or A ⊂ Λ^{2}V be a constant rank r subspace, dim A ≥ 2. Then E is a uniform vector bundle of splitting type O_{ℙ^{1}}^{r/2} ⊕ O_{ℙ^{1}}^{r/2}(1). In particular, r is even.

(b) If r is odd then c(r, m) = λ(r, m) = 1.

Proof. (a) The proof is by the same argument as in the proof of Theorem 6.14 (b).

(b) This follows from (a). However, in the symmetric case it was also known classically as a consequence of the Kronecker–Weierstrass theory giving a normal form for pencils of symmetric matrices of bounded rank, see [Gan, HP, Me]. ⊓ ⊔ Theorem 10.33 ([E1, IL]) A smooth n-dimensional projective variety X ⊂ ℙ^{N} with positive defect def X determines a (def X + 1)-dimensional linear subspace A ⊂ S^{2}ℂ^{N−1−def X} with associated short exact sequence

0 → N_{X}^{∗}H(1)|_{L} → O_{L} ⊗ ℂ^{N−1−def X} → N_{L}X → 0,

where H ∈ X^{∗}_{sm} and L = Sing X ∩ H is the contact locus.

Proof. Recall that L is a projective subspace of dimension def X. We have the following exact sequence

0 → N_{L}X → N_{L}H → N_{X}H|_{L} → 0

which holds because H is tangent to X along L. Dualising and twisting by O_{L}(1) we get

0 → N_{X}^{∗}H(1)|_{L} → N_{L}^{∗}H(1) → N_{L}^{∗}X(1) → 0.

It remains to use the fact that N_{L}H ≃ O_{L}(1) ⊗ ℂ^{N−1−def X} and, most importantly, that N_{L}X ≃ N_{L}^{∗}X(1) by Theorem 6.14 (a). ⊓ ⊔

The following theorem is a consequence of the general theory of uniform vector bundles [Sa].

Theorem 10.34 ([E1, IL])

(a) Suppose that A ⊂ S^{2}V is a linear (l+1)-dimensional subspace of constant rank r. Let 0 → K → O_{ℙ(A)} ⊗ V^{∗} → E → 0 be the corresponding exact sequence of vector bundles. If r ≤ l then E ≃ O_{ℙ(A)}^{r/2} ⊕ O_{ℙ(A)}^{r/2}(1) unless r = l = 2 and E = T ℙ^{2}(−1).

(b) Suppose that X ⊂ ℙ^{N} is a smooth projective variety. Let L be a contact locus with a generic tangent hyperplane. If def X ≥ N/2, then

N_{L}X = O_{L}^{(dim X−def X)/2} ⊕ O_{L}^{(dim X−def X)/2}(1).

Proof. (a) Indeed, E is uniform by Theorem 10.32 (a). Therefore, if r ≤ l then E either splits into a direct sum of line bundles or E is isomorphic to T ℙ^{l}(a) or T^{∗}ℙ^{l}(a) by [Sa]. If E splits then E ≃ O_{ℙ(A)}^{r/2} ⊕ O_{ℙ(A)}^{r/2}(1) by Theorem 10.32 (a). Suppose now that E is isomorphic to T ℙ^{l}(a) or T^{∗}ℙ^{l}(a). Since E ≃ E^{∗}(1), the first Chern class c_{1}(E) is equal to (l/2)H, where H is the class of the hyperplane section. Taking first Chern classes we conclude that E is isomorphic either to T ℙ^{2}(−1) or to T^{∗}ℙ^{2}(2), but these last two bundles are isomorphic.

(b) Let dim X = n and def X = k. By (a) and by Theorem10.33 we see that N_{L}X is isomorphic to either O_{L}^{(n−k)/2} ⊕ O_{L}^{(n−k)/2}(1) or T ℙ^{2}(−1). However, in the last case n = 4, k = 2, therefore X is a projective bundle ℙ_{C}(F) over a curve C by Example 6.17. Then L is embedded as a 2-plane in a fibre f of ℙ_{C}(F). Consider the exact sequence

0 → N_{L}f → N_{L}X → N_{f}X|_{L} → 0.

Since N_{L}f = O_{L}(1) and N_{f}X = O_{f}, this exact sequence reduces to

0 → O_{L}(1) → N_{L}X → O_{L} → 0.

It follows that N_{L}X = O_{L} ⊕ O_{L}(1). ⊓ ⊔

The following theorem was proved in [W]. Weaker results (but similar arguments) first appeared in [Sy].

Theorem 10.35 ([W]) Suppose 2 ≤ r ≤ m ≤ n. Then

(a) l(r, m, n) ≤ m + n − 2r + 1.

(b) l(r, m, n) = n − r + 1 if n − r + 1 does not divide (m − 1)!/(r − 1)!.

(c) l(r, r + 1, 2r − 1) = r + 1.

Sketch of the proof. Let A ⊂ Hom(V, W) be an l-dimensional linear subspace of constant rank r, dim V = m, dim W = n, m ≥ n. Let

0 → K → O_{ℙ(A)} ⊗ V → E → 0 and 0 → E → O_{ℙ(A)}(1) ⊗ W → N → 0

be the corresponding exact sequences of vector bundles. Then we have the following equations of Chern classes c(K)c(E) = 1 and c(E)c(N) = (1+H)^{n}, where H is the class of the hyperplane section. Thus, c(K)(1 + H)^{n} = c(N). If n − r + 1 ≤ i ≤ l − 1 then c_{i}(N) = 0 and looking at the coefficient of H^{i} we get

∑_{j=0}^{m−r} \binom{n}{i − j}k_{j} = 0,

where c_{j}(K) = k_{j}H^{j} and we use the convention that \binom{n}{i} = 0 if i < 0 or i > n. The coefficient matrix of this system of linear equations is

(\binom{n}{i − j})_{0≤j≤m−r, n−r+1≤i≤l−1} .

If l = m + n − 2r + 2 then this is square invertible matrix with determinant ∏_{j=0}^{m−r} j!. Thus k_{0} = 0 which is a contradiction since k_{0} = 1. This proves (a).

Proofs of (b) and (c) can be found in [W]. Notice that by Proposition 10.31 (a) we have l(r, m, n) ≥ n − r + 1, so only the opposite inequality should be proved. ⊓ ⊔

Another possible way of proving Theorem 10.35 (a) is to use the following theorem of R. Lazarsfeld (its application is immediate):

Theorem 10.36 ([La1]) Let X be l-dimensional projective variety. Let E and F be vector bundles of ranks m and n, respectively. Suppose that E^{∗} ⊗ F is ample and there is a constant rank r vector bundle map E → F. Then l ≤ m + n − 2r.

The (partial) symmetric analogue of this theorem was found in [IL]. Theorems 10.36 and 10.37 (and their proofs) are analogous to results related to the problem of the non-emptiness and the connectedness of degeneracy loci, see [La1, FL1, HT, HT1, Tu, IL].

Theorem 10.37 ([IL]) Let X be a smooth simply connected m-dimensional variety. Let E be a rank n vector bundle on X, and L a line bundle on X. Suppose that S^{2}E^{∗} ⊗L is an ample vector bundle and that there is a constant even rank r symmetric bundle map E → E^{∗} ⊗ L. Then m ≤ n − r.

Proof. Consider the projective bundle π : ℙ(E) → X. The symmetric bundle map E → E^{∗} ⊗ L defines a section s ∈ H^{0}(X, S^{2}E^{∗} ⊗ L) which pulls back to give a section π^{∗}s ∈ H^{0}(ℙ(E), π^{∗}(S^{2}E^{∗} ⊗ L)). There is a natural map π^{∗}π_{∗}O_{ℙ(E)}(2) → O_{ℙ(E)}(2). Since π_{∗}O_{ℙ(E)}(2) ≃ S^{2}E^{∗} this gives after tensoring with π^{∗}L a map u : π^{∗}(S^{2}E^{∗} ⊗ L) → O_{ℙ(E)}(2) ⊗ π^{∗}L. Let t = u ∘ π^{∗}s ∈ H^{0}(ℙ(E), O_{ℙ(E)}(2) ⊗ π^{∗}L). Let Y ⊂ ℙ(E) be the zero locus of t.

Let x ∈ X. The section t at [v] ∈ ℙ(E(x)) is the linear map λ → s(x)(v, v)λ and so vanishes if and only if s(x)(v, v) = 0. But by the hypothesis, this defines a rank r quadric hypersurface in ℙ(E(x)). Therefore, the fiber of π_{Y} : Y → X over each x ∈ X is a rank r quadric hypersurface in ℙ(E(x)).

We claim that O_{ℙ(E)}(2)⊗π^{∗}(L) is ample on ℙ(E). Indeed, since S^{2}E^{∗}⊗L is ample on X, O_{ℙ(S^{2}E^{∗}⊗L)}(1) is ample on ℙ(S^{2}E^{∗}⊗L). Clearly, ℙ(S^{2}E^{∗}⊗L) ≃ ℙ(S^{2}E^{∗}) and O_{ℙ(S^{2}E^{∗}⊗L)}(1) ≃ O_{ℙ(S^{2}E^{∗})}(1) ⊗ σ^{∗}L, where σ : ℙ(S^{2}E^{∗}) → X is the natural projection. Therefore, O_{ℙ(S^{2}E^{∗})}(1)⊗σ^{∗}L is ample on ℙ(S^{2}E^{∗}). The second Veronese map gives an inclusion i : ℙ(E) ⊂ ℙ(S^{2}E^{∗}) such that π = i ∘ σ. Then i^{∗}O_{ℙ(S^{2}E^{∗})}(1) = O_{ℙ(E)}(2). So i^{∗}(O_{ℙ(S^{2}E^{∗})}(1) ⊗ σ^{∗}L) ≃ O_{ℙ(E)}(2) ⊗ π^{∗}L is ample on ℙ(E) as required. In particular, ℙ(E) \ Y is an affine variety and its homology vanishes above its middle dimension: H_{i}(ℙ(E) \ Y, ℂ) = 0 for i ≥ m + n.

Let K denote the kernel and F denote the image of the map E → E^{∗} ⊗L. Then since the map has constant rank r, K and F are vector bundles on X of ranks n−r and r respectively. Since the map is symmetric, there is a symmetric isomorphism F ≃ F^{∗} ⊗L. We have a natural map p : ℙ(E)\ℙ(K) → ℙ(F) given on the fibers by the linear projection centered at ℙ(K(x)) ⊂ ℙ(E(x)). Then p is a ℂ^{n−r}-fiber bundle.

The isomorphism F ≃ F ⊗ L determines a hypersurface Z ⊂ ℙ(F) such that if ρ : ℙ(F) → X is the natural projection then the fiber of ρ|_{Z} : Z → X over each x ∈ X is a smooth quadric in ℙ(F (x)). Now ℙ(K) ⊂ Y and so p restricts to a ℂ^{n−r}-fiber bundle ℙ(E) \ Y → ℙ(F) \ Z. Thus, H_{i}(ℙ(E) \ Y, ℂ) ≃ H_{i}(ℙ(F) \ Z, ℂ) and by Lefschetz duality this is isomorphic to H^{2(m+r−1)−i}(ℙ(F), Z, ℂ). Hence H^{i}(ℙ(F), Z, ℂ) = 0 for i ≤ 2r + m − n − 2 and then using the long exact sequence of the pair (ℙ(F), Z) we conclude that H^{i}(Z, ℂ) ≃ H^{i}(ℙ(F), ℂ) for i ≤ 2r + m − n − 3.

Both bundles ρ : ℙ(F) → X and ρ|_{Z} : Z → X have smooth fibers and are defined over smooth simply-connected variety. Therefore, by Deligne’ theorem [GH1], the Leray spectral sequence for ρ and ρ|_{Z} degenerates at the E_{2} term. It follows that H^{∗}(ℙ(F), ℂ) = H^{∗}(X, ℂ) ⊗ H^{∗}(ℙ^{r−1}, ℂ) and H^{∗}(Z, ℂ) = H^{∗}(X, ℂ) ⊗ H^{∗}(Q, ℂ), where Q ⊂ ℙ^{r−1} is a smooth quadric. Let b_{j}(B) = dim H^{j}(B, ℂ be the Betti numbers. Then it is well-known that

b_{i}(ℙ^{r−1}) = \{ \begin{matrix}1 & for even i, 0 ≤ i ≤ 2(r − 1) \\ 0 & otherwise. \end{matrix}

Since r is even, we have

b_{i}(Q) = \{ \begin{matrix}1 & for even i, 0 ≤ i ≤ 2(r − 2) \\ 2 & i = r − 2 \\ 0 & otherwise. \end{matrix}

It follows that b_{r−2}(ℙ(F) ≠ b_{r−2}(Z). Since b_{i}(ℙ(F) = b_{i}(Z) for i ≤ 2r + m − n − 3, we finally obtain r − 2 > 2r + m − n − 3. Equivalently, m ≤ n − r. ⊓ ⊔

As an easy application of Theorem 10.37 we have the following theorem.

Theorem 10.38 ([IL]) Let V be an m-dimensional vector space, let r > 0 be an even number

max{dim A | A ⊂ ℙ(S^{2}V^{∗}) is of constant rank r} = m − r.

Example 10.39 Theorems 10.28 and 10.38 combined furnish a new proof of the formula dim X^{∗} ≥ dim X (Theorem 5.13 (b)).

References

[Al] P. Aluffi, Singular schemes of hypersurfaces, Duke Math. J., 80 (1995), 325–351

[AC] P. Aluffi, F. Cukierman, Multiplicities of discriminants, Manuscripta Math., 78 (1993), 245–258

[An] A.Z. Anan’in, Quasiderivations of the algebra of diagonal matrices, Tret’ya Sibirskaya shkola po algebre i analizu, Irkutsk University, 1990, 8–15 (in Russian)

[AVE] E.M. Andreev, E.B. Vinberg, A.G. Elashvili, Orbits of greatest dimension in semisimple linear Lie groups, Functional Anal. Appl., 1 (1967), 257–261

[ACGH] E. Arbarello, M. Cornalba, P.A. Griffiths, J. Harris, Geometry of Algebraic Curves, Vol. 1, Springer–Verlag, New York, 1985

[Ar] V.I. Arnold, Mathematical Methods of Classical Mechanics, Springer–Verlag, New York–Heidelberg–Berlin, 1978

[At] M.D. Atkinson, Primitive spaces of matrices of bounded rank, II, Jour. Australian Math. Soc., 34 (1983), 306–315

[AW] M.D. Atkinson, R. Westwik, Spaces of linear transformations of equal rank, Linear and Multilinear Algebra, 13 (1983), 231–239

[BCa1] P. Bala, R.W. Carter, Classes of unipotent elements in simple algebras. I, Math. Proc. Camb. Phil. Soc., 79 (1976), 401–425

[BCa2] P. Bala, R.W. Carter, Classes of unipotent elements in simple algebras. II, Math. Proc. Camb. Phil. Soc., 80 (1976), 1–18

[BC] E. Ballico, L. Chiantini, On smooth subcanonical varieties of codimension 2 in ℙ^{n}, n ≥ 4, Ann. Mat. Pure Appl., 135:4 (1983), 99-117

[Bar] F. Bardelli, The Sylvester pentahedron and some applications to moduli spaces, preprint

[Ba] W. Barth, Transplanting cohomology class in complex projective space, Amer. J. Math, 92 (1970), 951–961

[Be] L.B. Beasley, Null spaces of matrices of bounded rank, Current Trends in Matrix Theory (R. Grone and F. Uhlig, Eds.), North–Holland, Amsterdam, 1987, 45–50

[Bea] A. Beauville, Fano contact manifolds and nilpotent orbits, Comm. Math. Helv., 73:4 (1998), 566–583

[Bek] N.D. Beklemishev, Classification of quaternary cubic forms not in general position, In: Voprosy teorii grupp i gomologicheskoy algebry, Yaroslavl, 1981 (in Russian)

[BFS1] M.C. Beltrametti, M.L. Fania, A.J. Sommese, On the discriminant variety of a projective manifold, Forum Math, 4 (1992), 529–547

[BFS2] M.C. Beltrametti, M.L. Fania, A.J. Sommese, On the adjunction theoretic classification of projective varieties, Math. Ann., 290 (1991), 31–62

[BS] M.C. Beltrametti, A.J. Sommese, The adjunction theory of complex projective varieties, Expositions in Math., 16 (1995), W. de Gruyter, Berlin [BSW] M.C. Beltrametti, A.J. Sommese, J.A. Wiśniewski, Results on varieties

with many lines and their applications to adjunction theory, Proceedings of the Bayreuth Conference “Complex Algebraic Varieties”, April 1990. Lecture Notes in Math. 1507, Springer, 1992, 16–38

[Berg] G.M. Bergman, The diamond lemma for ring theory, Adv. in Math., 29/2 (1978), 178–218

[BGG] I.N. Bernstein, I.M. Gelfand, S.I. Gelfand, Schubert cells and the cohomology of G/P, Uspechi Matematicheskih Nauk, 28/3 (1973), pp. 3–26

[Ber] E. Bertini, Introduzione alla geometria proiettriva degli iperspazi, Enrico Spaerri, Piza, 1907

[Bot] R. Bott, Homogeneous vector bundles, Ann. Math., 66/2 (1957), pp. 203–248 [Bo] N. Bourbaki, Groupes et algébres de Lie, Hermann, Paris 1968

[Br] M. Brion, Représentations exceptionnelles des groupes semi-simples, Ann. Sci. Ec. Norm. Sup., 18 (1985), 345–387

[Ca] A. Cayley, On the theory of elimination, Collected papers Vol. 1, Cambridge Univ. Press, 1889, pp. 370–374

[Ca1] A. Cayley, On the theory of linear transformations, Collected papers Vol. 1, Cambridge Univ. Press, 1889, pp. 80–94

[Ca2] A. Cayley, On linear transformations, Collected papers Vol. 1, Cambridge Univ. Press, 1889, pp. 95–112

[CR] T. Cecil, P. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc., 269 (1982)

[CKM] H. Clemens, J. Kollar, S. Mori, Higher dimensional complex geometry, Asterisque, 166 (1988)

[CM] D. Collingwood, W. McGovern, Nilpotent orbits in semisimple Lie algebras, Van Nostrand Reinhold Company, 1993

[CW] C. De Concini, J. Weiman, A formula with nonnegative terms for the degree of the dual variety of a homogeneous space, Proc. AMS, 125:1 (1997), 1–8

[De] L. Degoli, Due Nouvi Teoremi Sui Sistemi Lineari Di Quadriche A Jacobiana Identicamente Nulla, Collect. Math., 35 (1984), 125–139

[Del] P. Deligne, Le déterminant de la cohomologie, In: Current trends in arithmetic algebraic geometry, Contemporary Math., No. 67,Amer. Math. Soc., 1987, pp. 93–177

[DK] P. Deligne, N.Katz, Groupes de monodromie en geometrie algebrique, Lecture Notes in Math. 340, Springer, 1973

[Di1] A. Dimca, Milnor numbers and multiplicities of dual varieties, Rev. Roumaine Math. Pures (1986), 535–538

[Di2] A. Dimca, Topics on real and complex singularities, Vieweg Advanced Lectures (1987)

[D] R. Donagi, On the geometry of Grassmanians, Duke Math. J., 44 (1977), 795– 837

[E1] L. Ein, Varieties with small dual varieties, I, Invent. Math., 86 (1986), 63–74 [E2] L. Ein, Varieties with small dual varieties, II, Duke. Math. J., 52 (1985),

895–907

[E3] L. Ein, Stable vector bundles on projective space in char p > 0, Math. Ann., 254 (1980), 53–72

[E4] L. Ein, The ramification divisors for branched coverings of ℙ_{k}^{n}, Math. Ann., 261 (1982), 483–485

[ES] L. Ein, N. Shepherd–Barron, Some special Cremona transformations, Amer. J. Math., 111 (1989), 783–800

[EH] D. Eisenbud, J. Harris, Vector spaces of matrices of low rank, Adv. Math., 70 (1988), 135–155 [El1] A.G. Elashvili, Canonical form and stationary subalgebras of points of general

position for simple linear Lie groups, Functional Anal. Appl., 6 (1972), 44–53 [El2] A.G. Elashvili, Stationary subalgebras of points of general position for irre-

ducible linear Lie groups, Functional Anal. Appl., 6 (1972), 139–148

[EHS] G. Elencwajg, A. Hirschowitz, M. Schneider, Les fibres uniformes de rang au plus n sur ℙ_{n}(ℂ), Proceedings of the Nice Conference 1979 on Vector bundles and Differential equations

[Fa] B. Fantechi, On the superadditivity of secant defects, Bull. Soc. Math. France, 118 (1990), 85–100

[Fi] E. Fisher, Über die Cayleyshe Eliminationsmethode, Math. Zeit., 26 (1927), 497–550

[Fra] W. Franz, Die torsion einer Überdeckung, J. Reine u. angew. Math., 173 (1935), 245–254

[Fr] H. Freudenthal, Beziehungen der E_{7} und E_{8} zur Oktavenebene I-XI, Nederl. Akad. Wet., Proc., 57 (1954), 218–230, 363–368, 58 (1955), 151–157, 277–285, 62 (1959), 165–179, 180–191, 192–201, 447–465, 466–474, 66 (1963), 457–471, 472–487

[Fr1] H. Freudenthal, Lie groups in the foundation of geometry, Adv. Math., 1 (1964), 145–190

[Fub] G. Fubini, Il problema della deformazione proiettiva delle ipersuperficie, Rend. Acad. Naz. dei Lincei, 27 (1918), 147–155

[F] T. Fujita, Projective threefolds with small secant varieties, Sci. Papers College of General Education. Univ. Tokio, 32:1 (1982), 33–46

[FR] T. Fujita, J. Roberts, Varieties with small secant varieties: the extremal case, Amer. J. Math., 103 (1981), 953–976

[Fu1] W. Fulton, Intersection Theory, Ergebnisse der Mathematik 2, Springer, 1984 [Fu2] W. Fulton, On the topology of algebraic varieties, Proc. Symp. Pure Math.,

46 (1987), 15–46

[FH] W. Fulton, J. Hansen, A connectedness theorem for projective varieties, with applications, Ann. Math., 110:1 (1979), 159–166

[FL] W. Fulton, R. Lazarsfeld, Connectivity and its applications in algebraic geometry, Proc. Midwest Algebraic Geometry Conf. Univ. Illinois, 1980. Lect. Notes Math. Springer–Verlag, Berlin–Heidelberg–New York, 862 (1981), 26–92

[FL1] W. Fulton, R. Lazarsfeld, On the connectedness of degeneracy loci and special divisors, Acta Math., 146 (1981), 271–283

[Ga] P. Gabriel, Unzerlegbare Darstellungen, I, Manuscripta Math., 6 (1972), 71– 103

[Gan] F.R. Gantmacher, The theory of matrices Vol.2, Chelsea Publishing Company, New York, 1959

[GK] I.M. Gelfand, M.M. Kapranov, On the dimension and degree of projectively dual varieties: a q-analogue of the Katz–Kleiman formula, in: Gelfand Mathematical Seminars 1990–1992, Birkhaüser 1993, pp. 27–34

[GKZ1] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinski, A-discriminants and the Cayley–Koszul complexes, Dokl. Akad. Nauk SSSR, 307 (1989), 1307–1311 (Russian)

[GKZ2] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinski, Discriminants, resultants, and multidimensional determinants, Birkhauser, Boston, 1994

[GKZ3] I.M. Gelfand, M.M. Kapranov, A.V. Zelevinski, Hyperdeterminants, Adv. Math., 84 (1990), pp. 237–254

[Go] N. Goldstein, Ampleness and connectedness in complex G/P, Trans. AMS, 274 (1982), 361–374

[GR] H. Grauert, O. Rimenschneider, Verschwindungssätze für Annalische Kogomologiegruppen auf Komplexen Räumen, Inv. Math., 11 (1970), 263–292 [GH] P. Griffiths, J. Harris, Algebraic geometry and local differential geometry,

Ann. Sci. Ec. Norm. Super., 12 (1979), 355–432

[GH1] P. Griffiths, J. Harris, Principles of Algebraic Geometry, Wiley–Interscience, London–New–York, 1979

[HT] J. Harris, L.W. Tu, The connectedness of symmetric degeneracy loci: odd ranks, Topics in Algebra: Banach Center Publications, Vol. 26, Part 2, PWN-Polish Scientific Publishers, Warsaw, 1990, pp. 249–256

[HT1] J. Harris, L.W. Tu, On symmetric and skew-symmetric determinantal varieties, Topology, 23 (1984), 71–84

[Ha1] R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer–Verlag, New York–Heidelberg–Berlin, 1977

[Ha2] R. Hartshorne, Varieties of low codimension in projective space, Bull. Amer. Math. Soc., 80 (1974), 1017–1032

[Ha3] R. Hartshorne, Ample vector bundles, IHES Publ. Math., 29 (1966), 63–94 [HK] A. Hefez, S. Kleiman, Notes on duality for projective varieties

[HRo] L. Hille, G. Röhrle, A classification of parabolic subgroups of classical groups with a finite number of orbits on the unipotent radical, Transformation Groups, 4:1 (1999), 35–52

[Hir] F. Hirzebruch, Topological Methods in Algebraic Geometry, Grund. Math. Wiss. 131, 1966, Springer–Verlag

[HP] W.V.D. Hodge, D. Pedoe, Methods of Algebraic Geometry, Cambridge University Press, Cambridge, 1994

[Ho1] A. Holme, On the dual of the smooth variety, Lect. Notes in Math., 732 (1979), Springer–Verlag, 144–156

[Ho2] A. Holme, The geometric and numerical properties of duality in projective algebraic geometry, Manuscripta Math., 61 (1988), 145–162

[HR] A. Holme, J. Roberts, Pinch points and multiple locus for generic projections of singular varieties, Adv. Math., 33 (1979), 212–256

[IL] B. Ilic, J.M. Landsberg, On symmetric degeneracy loci, spaces of symmetric matrices of constant rank and dual varieties, Math. Ann., 314 (1999), 159–174

[I] P. Ionescu, An enumeration of all smooth projective varieties of degree 5 and 6, Increst Preprint Series Math., 74 (1981)

[J] N. Jacobson, Structure and representations of Jordan algebras, Publ. Amer. Math. Soc., 39 (1968)

[KOY] H. Kaji, M. Ohno, O. Yasukura, Adjoint varieties and their secant varieties, Indag. Math., 10 (1999), 45–57

[Kan] I.L. Kantor, Models of exceptional Lie algebras, Soviet Math. Dokl., 14 (1973), 254–258

[Ka] N. Katz, Pinceaux de Lefschetz; Theoreme d’existence, in: SGA 7, Lecture Notes in Math., 340, 212–253

[KMM] Y. Kawamata, K. Matsuda, K. Matsuki, Introduction to the minimal model program, Algebraic Geometry, Sendai 1985. Adv. Stud. Pure Math., 10 (1987), 283–360

[Kaw] Y. Kawamata, The cone of curves of algebraic varieties, Ann. Math, 119 (1984), 603–633

[Ke] G. Kempf, On the collapsing of homogeneous bundles, Invent. Math. 37 (1976), 229–239

[Kl1] S. Kleiman, Enumerative theory of singularities, Real and Complex Singularities (Proc. Ninth Nordic Summer School/NAVF Sympos. Math., Oslo, 1976), 297–396

[Kl2] S. Kleiman, About the conormal scheme, “Complete intersections”, Proceedings of Acireale (1983), Lecture Notes in Math. 1092, Springer, 1984, 161–197 [Kl3] S. Kleiman, Tangency and duality, Proceedings of the 1984 Vancouver Confer-

ence in Algebraic Geometry, July 2–12, 1984, ed. by J. Carrell, A.V. Geramita, and R. Russell. Canadian Math. Soc. Conference Proceedings 6, (1986), 163–225

[Kl4] S. Kleiman, Concerning the dual variety, Proc. 18th Scand. Congr. Math. Aarhus, 1981, Birkhäuser Verlag, Basel–Boston–Stuttgart, 386–396

[Kl5] S. Kleiman, Plane forms and multiple point formulas, Algebraic threefolds (Varenna, 1981), pp. 287–310, Lecture Notes in Math., 947 (1982)

[Kl6] S. Kleiman, Toward a numerical theory of ampleness, Ann. Math, 84 (1966), 293–344

[KZ] K. Knight, A. Zelevinsky, Representations of quivers of type A and the multisegment duality, Adv. Math, 117 (1996), 273–293

[KM] F. Knop, G. Mentzel, Duale Varietäten von Fahnenvarietäten, Comm. Math. Helv., 62 (1987), 38–61

[KnMu] F. Knudsen, D. Mumford, Projectivity of moduli spaces of stable curves I. Preliminaries on Det and Div, Math. Scand., 39 (1976), 19–55

[KO] S. Kobayashi, T. Ochiai Characterization of complex projective spaces and hyperquadrics, J. Math. Kyoto Univ., 13:1 (1973), 31–47

[Ko] Y. Kollar, The structure of algebraic threefolds. An introduction to Mori’s program, Bull. AMS, 17:2 (1987), 211–273

[KS] A. Kumpera, D. Spencer, Lie Equations, Vol. 1: General Theory, Ann. of Math. Stud., 73 (1972), Princeton University Press

[Lam] I. Lamotke, The topology of complex projective varieties after S. Lefschetz, Topology, 20 (1981), 15–51

[Land] A. Landman, Examples of varieties with small dual varieties. Picard– Lefschetz theory and dual varieties, Two lectures at Aarhus Univ., (1976)

[Lan] J.M. Landsberg, On degenerate secant and tangential varieties and local differential geometry, Duke Math. J., 85 (1996), 605–634

[Lan1] J.M. Landsberg, Differential–geometric characterization of complete intersections, J. of Diff. Geom., 44 (1996), 32–73

[Lan2] J.M. Landsberg, On second fundamental form of projective varieties, Invent. Math., 117 (1994), 303–315

[Lan3] J.M. Landsberg, On degenerate secant and tangential varieties and local differential geometry, Duke Math. J., 85 (1996), 1–30

[Lan4] J.M. Landsberg, Algebraic geometry and projective differential geometry, preprint, (1998)

[LM] J.M. Landsberg, L. Manivel, On the projective geometry of homogeneous varieties, preprint, (2000)

[LP] A. Lanteri, M. Palleschi, Projective manifolds containing many rational curves, Indiana Math. J., 36 (1987), 857–865

[LPS] A. Lanteri, M. Palleschi, A.J. Sommese, Discriminant loci of varieties with smooth normalization, preprint (1998)

[LS] A. Lanteri, D. Struppa, Projective 7-folds with positive defect, Compositio Math., 61 (1987), 329–337

[L] M.E. Larsen, On the topology of complex projective manifolds, Invent. Math., 19 (1972), 251–260

[Las] A. Lascoux, Degree of the dual Grassman variety, Comm. Algebra, 9/11 (1981), 1215–1225

[La] R. Lazarsfeld, An example of 16-dimensional projective variety with a 25dimensional secant variety, Math. Letters, 7 (1981), 1–4

[La1] R. Lazarsfeld, Some applications of the theory of positive vector bundles, LNM 1092, Complete Intersections (Acireale 1983), Springer–Verlag, Berlin, 1984, pp. 29–61 [LV] R. Lazarsfeld, A. Van de Ven, Topics in the Geometry of Projective Space,

Recent Work of F.L. Zak, DMV Seminar, Birkhauser, 1984

[Li] W. Lichtenstein, A system of quadrics describing the orbit of the highest weight vector, Proc. A.M.S., 84 (1982), 605–608

[Ma] Yu.I. Manin, Cubic forms. Algebra, geometry, arithmetic, M., Nauka, 1972. English transl.: North–Holland, Amsterdam–London 1974

[MM] H. Matsumura, P. Monsky, On the automorphisms of hypersurfaces, J. Math. Kyoto Univ., 3:3 (1964), 347–361

[Me] R. Meshulam, On two-parameter families of symmetric matrices, Lin. Alg. Appl., 216 (1995), 93–96

[Mi] J. Milnor, Singular points of complex hypersurfaces, Ann. of Math. Studies 61, Princeton Univ. Press (1968)

[MW] C. Moeglin, J.L. Waldspurger, Sur l’involution de Zelevinski, J. Reine Angew. Math, 372 (1986), 136–177

[M] B.G. Moishezon, Algebraic homology classes of algebraic varieties, Izvestiya AN SSSR, 31 (1967) (in Russian)

[Mo] E.H. Moore, General Analysis – Part I, Mem. Amer. Phil. Soc., 1 (1935)

[Mor] S. Mori, Threefolds whose canonical bundle is not numerically effective, Ann. Math., 116 (1982), 133–176

[Mor1] S. Mori, Projective manifolds with ample tangent bundle, Ann. Math., 110 (1979), 593–606

[Mu] S. Mukai, New classification of Fano threefolds and Fano manifolds of coindex 3, preprint (1988)

[MRS] I. Muller, H. Rubenthaler, G. Schiffmann, Structure des espaces préhomogènes associés à certaines algébres de Lie graduées, Math. Ann., 274 (1986), 95–123

[Mum] D. Mumford, Some footnote of the work of C.P. Ramanujam, in: C.P. Ramanujam, a Tribute, pp. 247–262, Springer, 1978

[Mum1] D. Mumford, Complex projective varieties, Springer–Verlag, Berlin–Heidelberg–New York, 1979

[MFK] D. Mumford, J. Fogarty, F. Kirwan, Geometric Invariant Theory, Ergebnisse der Mathematik und Grenzgebiete, vol. 34, Springer–Verlag, New York– Heidelberg–Berlin, 1994. (3rd edition)

[N] A. Némethi, Lefschetz theory for complex hypersurfaces, Rev. Roumaine Math. Pures (1988), 233–250

[OSS] C. Okonek, H. Spindler, M. Schneider, Vector bundles on complex projective spaces, Prog. Math. 3, Basel–Boston, Birkhäuser, 1980

[OV] A.L. Onischik, E.B. Vinberg, Lie groups and Lie algebras III, Encyclopedia of Mathematical Sciences 41, Springer–Verlag 1994

[Pan] D.I. Panyushev, Parabolic subgroups with abelian unipotent radical as a testing site for invariant theory, Canad. J. Math., 51:3 (1999), 616–635

[Pa1] A. Parusiǹski, A generalization of the Milnor number, Math. Ann., 281 (1988), 247–254

[Pa2] A. Parusiǹski, Multiplicity of the dual variety, Bull. London Math. Soc., 23 (1991), 428–436

[Pau] C. Pauly, Self-duality of Coble’s quartic hypersurface and applications, preprint

[Pe] R.A. Penrose, A generalized inverse for matrices, Proc. Camb. Phil. Soc., 51 (1955)

[P] R. Piene, Plücker formulas, Doctoral thesis, M.I.T. Cambridge, MA, 1976 [Po] S. Poljak, Maximum rank of powers of a matrix of a given pattern, Proc.

Amer. Math. Soc., 106:4 (1989), 1137–1144 [Pop] V. Popov, Self-dual algebraic varieties and nilpotent orbits, Proceedings of the

International Colloquium “Algebra, Arithmetic, and Geometry”, Tata Institute of fundamental Research, Bombay, 2000

[PR] V. Popov, G. Röhrle, On the number of orbits of a parabolic subgroup on its unipotent radical, Algebraic Groups and Lie Groups, Lehrer G.I., Ed., Austr. Math. Soc. Lecture Series, 9 (1997)

[PV] V.L. Popov, E.B. Vinberg, Invariant Theory, in: Algebraic Geometry IV, Encyclopedia of Mathematical Sciences, Springer–Verlag, 55 (1994), 123–278

[Py] V.S. Pyasetskii, Linear Lie groups acting with finitely many orbits, Funct. Anal. Appl. 9 (1975), 351–353

[Q] D. Quillen, Determinants of Cauchy–Riemann operators on Riemann surfaces, Funct. Anal. Appl., 19 (1985), 31–34

[RS] N. Ray, I. Singer, R-torsion and the Laplacian for Riemannian manifolds, Adv. Math., 7 (1971), 145–210

[Ra1] Z. Ran, The (dimension+2)-secant Lemma, Invent. Math., 106 (1991), 65–71 [Ra2] Z. Ran, On projective varieties of codimension two, Invent. Math., 73 (1983),

333-336

[Ra3] Z. Ran, The structure of Gauss-like maps, Compos. Math., 52 (1984), 171– 177

[Ri] R.W. Richardson, Finiteness theorems for orbits of algebraic groups, Indag. Math., 88 (1985), 337–344

[Ri1] R.W. Richardson, Conjugacy classes of n-tuples in Lie algebras and algebraic groups, Duke Math. J., 57 (1988), 1–35

[RRS] R.W. Richardson, G. Röhrle, R. Steinberg, Parabolic subgroups with abelian unipotent radical, Invent. Math., 110 (1992), 649–671

[Rob] J. Roberts, Generic projections of algebraic varieties, Amer. J. Math., 93 (1971), 191–214

[R] T. Room, A Synthesis of Clifford matrices and its generalization, Amer. J. Math., 74 (1952), 967–984

[Ro] B. Rosenfeld, Geometry of Lie groups, Mathematics and its Applications 393, Kluwer Academic Publishes 1997

[Ru] W. Rudin, Real and Complex Analysis, Mc Graw Hill, 1987

[Sa] E. Sato, Uniform vector bundles on a projective space, J. Math. Soc. Japan, 28 (1976), 123–132

[Schl] L. Schläffli, Gesammelte Abhandlungen, Band. 2, pp. 9–112L, Birkhäuser Verlag, Basel, 1953

[Se] B. Segre, Bertini forms and hessian matrices, L. London Math. Soc., 26 (1951), 164–176

[Sn] D. Snow, The nef value and defect of homogeneous line bundles, Trans. AMS, 340:1 (1993), 227–241

[Sn1] D. Snow, On the ampleness of homogeneous vector bundles, Trans. AMS, 294:2 (1986), 227–241

[S] A.J. Sommese, Hyperplane section of projection surface I — the adjunction mapping, Duke Math. J., 46 (1979), 377-401

[S1] A. Sommese, Submanifolds of abelian varieties, Math. Ann., 233 (1978), 229– 256

[Sy] J. Sylvester, On the dimension of spaces of linear transformations satisfying rank conditions, Linear Algebra and its Applications, 76 (1986), 1–10

[Ta] R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math., 10 (1973), 495–506

[Tan] H. Tango, Remark on varieties with small secant varieties, Bul. Kyoto Univ. Ed. Ser. B, 60 (1982), 1–10 [Te] Terracini, Alcune questioni sugli spazi tangenti e osculatori ad una varieta, I,

II, III, Societa dei Naturalisti e Matematici (1913), 214–247

[T1] E. Tevelev, Subalgebras and discriminants of anticommutative algebras, Izvestiya RAN, Ser. Mat., 63:3 (1999), 169–184

[T2] E. Tevelev, Generic algebras, Transformation Groups, 1 (1996), 127–151

[T3] E. Tevelev, Isotropic subspaces of multilinear forms, Matematicheskie Zametki, 2001

[T4] E. Tevelev, Moore–Penrose inverse, parabolic subgroups, and Jordan pairs, preprint, 2000

[T5] E. Tevelev, Generic algebras, PhD Thesis

[T6] E. Tevelev, Discriminants and quasi-derivations of commutative algebras, Uspechi Matematicheskih Nauk, 54:2 (1999), 189–190

[Tu] L.W. Tu, The connectedness of symmetric and skew symmetric degeneracy loci: even ranks, Trans. AMS, 313 (1989), 381–392

[V] A. Van de Ven, On the 2-connectedness of the very ample divisor on a surface, Duke Math. J., 46 (1979), 403–407

[Vi] E.B. Vinverg, The Weyl group of a graded Lie algebra, Math. USSR Izv., 10 (1976), 463–495

[Vi1] E.B. Vinberg, Generalized derivations of algebras, AMS Transl., 163 (1995), 185–188

[Wa] A.H. Wallace, Tangency and duality over arbitrary fields, Proc. London Math. Soc., 6:3 (1956), 321–342

[Wei] A. Weil, Foundations of algebraic geometry, A.M.S. Coloquium Publ., 29 (1962), Providence, R.I.

[W] R. Westwick, Spaces of matrices of fixed rank, Linear and Multilinear Algebra, 20 (1987), 171–174

[W1] R. Westwick, Spaces of matrices of fixed rank, II, Linear Algebra and its Applications, 235 (1996), 163–169

[We] J. Weyman, Calculating discriminants by higher direct images, Trans. Amer. Math. Soc., 343 (1994), 367–389

[WZ] J. Weyman, A. Zelevinsky, Singularities of Hyperdeterminants, Ann. Inst. Fourier, 46:3 (1996), 591–644

[WZ1] J. Weyman, A. Zelevinsky, Multiplicative properties of projectively dual varieties, Manuscripta Math., 82:2 (1994), 139–148

[Wi] P. Wilson, Towards birational classification of algebraic varieties, Bull. London Math. Soc., 19 (1987), 1–48

[Wis] J.A. Wiśniewski, Length of extremal rays and generalized adjunction, Math. Z., 200 (1989), 409–427

[Wis1] J.A. Wiśniewski, On a conjecture of Mukai, Manuscripta Math., 68 (1990), 135–141

[Z1] F.L. Zak, Projection of algebraic varieties, Mat. Sbornik, 116 (1981), 593–602 (in Russian). English Translation: Math. USSR Sbornik, 44 (1983), 535–544

[Z2] F.L. Zak, Severi varieties, Mat. Sbornik, 126 (1985), 115–132 (in Russian).

[Z3] F.L. Zak, The structure of Gauss maps, Funct. Anal. i Prilozh., 21 (1987), 39–50 (in Russian).

[Z4] F.L. Zak, Tangents and secants of algebraic varieties, Translations of Math. Monographs, 127, AMS 1993

[Z5] F.L. Zak, Linear systems of hyperplane sections of varieties of low codimension, Funct. Anal. i Prilozh., 19 (1985), 165–173

[Za] O. Zariski, Theory and applications of holomorphic functions on algebraic varieties over arbitrary ground fields, Mem. A.M.S. No. 5 (1951), Providence, R.I.

Index

1-cycle 134 ℙ^{d}-bundle 142 ℚ-divisors 135 μ-number 123

adjoint variety 29 admissible collection 46 Albert algebra 26, 39 ample 16 ample vector bundle 51 ampleness 52 annihilator 1, 154 associated hypersurfaces 88

base-point free 16 based exact complex 46 Beilinson spectral sequence 95 bigraded complex 50 bitangent 10 Borel subgroup 21 Borel–Weyl-Bott theorem 22 boundary format 110 Bourbaki numbering 21

Cartan Lemma 99 Cartier divisor 15 caustic 9 Cayley determinant 43 Cayley numbers 26 Cayley trick 53, 88 character group 21 characteristic of a nilpotent 64 Chern class 105 Chern polynomial 106 Chow forms 89 class formula 107, 122 class of a plane curve 10 classical discriminant 23, 48, 108, 118 classical resultant 53 Clifford algebra 25 cohomology of Koszul complexes 42 complete linear system 16 cone of effective 1-cycles 134 conical action 30 conical orbit 30 conormal bundle 4 conormal variety 2 contact locus 7, 89 contact scheme 129 Contraction Theorem 135 correspondence of branches 10 cotangent bundle 4 cusp 10 cuspidal point 10

Darboux frame 98

defect of a projective variety 3 defect of smooth divisors 17 degeneracy locus 176 degree of a cycle 105 Del Pezzo fibration 142 desingularization of a dual variety 6 determinant of a complex 45 determinant of a supervector space 43 determinant of a vector space 43 discriminant 3 discriminant complex 46 discriminant of a binary form 3 discriminant of a linear operator 27 discriminant of an element in a Lie

algebra 28 discriminant spectral sequence 49 dominant weight 21 doublings of linear systems 173 dual of the curve 7 dual of the smooth surface 92 dual plane curve 8 dual projective subspace 1 dual quiver 32 dual variety 2 dual variety of a complete intersection

52 dual variety of a curve 48 dual variety of a smooth complete

intersection 106 dual vector space 1

effective 1-cycle 134 effective divisor 15 Ein Theorems 89 embedded projective tangent space 2 Euler characteristic 119 extremal rational curve 135 extremal ray 135

Fano variety 137 Fermat curve 8 first jet 41 flex 10 Frobenius integrability condition 98 Fulton–Hansen connectedness theorem

71 fundamental form 97 fundamental forms of flag varieties

104 fundamental weights 21, 144

Gauss image 77 Gauss map 96 Grassmanian 23 Grassmanian fibration 88 group of norm similarities 39

half-spinor representation 25 Hartshorne conjecture 78, 170 height of a nilpotent orbit 64 Hessian matrix 81 higher fundamental forms 103 Hilbert polynomial 119 Hilbert scheme 91 homogeneous 𝔰𝔩_{2}-triple 62 homogeneous coordinates 1 homogeneous real hypersurface 170 horizontal 1-form 98 hyperdeterminants 87, 110, 130 hyperdeterminants, existence of 87 hyperplane 1

index of a Fano variety 137 inflection point 10 intersection product 134 isotropic Grassmanian 24 isotropic subspace 24

Jacobian ideal 122 jet bundle 41 join 75 Jordan algebra 26 Jordan decomposition 154

Katz dimension formula 81 Katz–Kleiman–Holme formula 106 Killing form 21 Kleiman criterion 135 Kobayashi–Ochiai characterisation

142 Kodaira dimension 92 Kostant partition function 32 Koszul complex 42 Kronecker–Weierstrass theory 174

Lagrangian variety 5 Landman formula 107 Legendre transformation 9 length of extremal rays 136 Levi subgroup 31 linear system of divisors 16 linear systems of bounded rank 172 linear systems of constant rank 172 linear systems of quadrics 172 linear systems of rank bounded below

172 linearly equivalent divisors 15 linearly normal variety 14 lines on flag varieties 104 local equations of a divisor 15 lower canonical string 36

matrix format 130 Matsumura-Monsky theorem 151 Maurer–Cartan equations 98 Maurer–Cartan forms 98 maximal parabolic subgroup 22 maximal torus 21 Meusnier–Euler theorem 99 Milnor class 123 Milnor number 123 moment map 31 Monotonicity theorem 95 Moore–Penrose inverse 61 Mori Cone Theorem 135 moving frames 98 Mukai fibration 142 multiple tangent 10 multiplicities of the discriminant 124 multisegment 32 multisegment duality 32

nef divisors 135 nef value 136 nef value morphism 136 negative roots 21 Neron–Severi group 134 non-degenerate variety 14 norm similarities 26 normal bundle 4 numerically equivalent 134 numerically trivial 134

octonions 26 ordinary double point 10 osculating sequence 99 osculating space 99, 103

parabolic subgroup 22 parity theorem 91, 129 partial order on the weight lattice 22 pencils of symmetric matrices 174 permanent 109 Picard number 134 Plücker embedding 23, 102 Plücker formulas 10 polarized variety 17, 116, 133 positive roots 21 Product theorem 84 projection from a subspace 12 projective second fundamental form

97 projective subspace 1 properties of jet bundles 41 Pyasetskii Theorem 30

quadratic nilpotent 153 quadric fibration 142 quasi-annihilator 154 quasiderivation 155 quotient projective space 12

ranks of a projective variety 108 Rationality Theorem 136 Reflexivity Theorem 4 regular algebra 154 Reidermeister torsion 43 relative secant variety 75 relative tangential variety 75 representations of quivers 32 resultant variety 52 Richardson Lemma 31 root lattice 144 ruled variety 7

Schläffli formula 11 scroll 142 secant variety 74 Segre class 106 Segre embedding 74, 84, 101 Segre Theorem 83 self-dual variety 168 semi-ample divisor 133 Severi variety 27, 78, 79, 102 short grading 34 simple roots 21 singular locus of hyperdeterminants

130 singular locus of the dual variety 130 singular scheme 122 singular subalgebra 153 smoothness principle 97 spectral sequence of the double complex

49 spinor variety 24, 102 stable point 151 stably twisted discriminant complex

47 standard projective space 1 strictly dominant weight 21 supervector space 43 Sylvester formula 48 symplectic structure 5 symplectic structure on the cotangent

bundle 5 symplectic variety 5

tangent bundle 4 tangent hyperplane 2 tangent star 75 tangential variety 74 Terracini lemma 75

uniform vector bundle 90, 174 upper canonical string 37

varieties with small dual varieties 142 variety of complexes 67 Veronese curve 3 Veronese embedding 23, 102 very ample 16 very ample vector bundle 52

weight lattice 21

weight of a multisegment 32 Weyman’s bigraded complex 50

Zak’s theorem on linear normality 77 Zak’s theorem on Severi varieties 78 Zak’s theorem on tangencies 76 Zelevinsky involution 32