Projective Duality and Homogeneous Spaces

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Encyclopaedia of Mathematical Sciences Volume 133 Invariant Theory and Algebraic Transformation Groups IV Subseries Editors: R.V. Gamkrelidze V.L. Popov

E. A. Tevelev

Projective Duality and Homogeneous Spaces

4y Springer

Author Evgueni A. Tevelev Department of Mathematics University of Texas at Austin 78712 Austin, Texas, USA e-mail: [email protected]

Founding editor of the Encyclopaedia of Mathematical Sciences: R. V. Gamkrelidze

Mathematics Subject Classification (2000): 14-XX

ISSN 0938-0396 ISBN 3-540-22898-5 Springer Berlin Heidelberg New York This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilm or in any other way, and storage in data banks. Duplication of this publication or parts thereof is permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer. Violations are liable for prosecution under the German Copyright Law. Springer is a part of Springer Science+Business Media GmbH springeronline.com ©Springer-Verlag Berlin Heidelberg 2005 Printed in Germany The use of general descriptive names, registered names, trademarks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. Typesetting: by the authors using a Springer iSTjiX macro package Production: LE-TgK Jelonek, Schmidt & VBckler GbR, Leipzig Cover Design: E. Kirchner, Heidelberg, Germany Printed on acid-free paper 46/3142 YL 5 4 3 2 1 0

Preface

During several centuries various reincarnations of projective duality have inspired research in algebraic and differential geometry, classical mechanics, invariant theory, combinatorics, etc. To put it simply, projective duality is the systematic way of recovering a projective variety from the set of its tangent hyperplanes. In this survey I have tried to emphasize that there are many different aspects of projective duality and that it can be studied using a wide range of methods. But at the same time I was pushing hard to minimize the technical details in the hope of writing a text that requires a knowledge of only basic facts from algebraic geometry and the theory of algebraic (or Lie) groups. Most proofs in this book are compilations from various sources. Projective duality is defined for arbitrary projective varieties and it does not seem natural a priori to consider varieties with symmetries. However, it turns out that many important examples carry the natural action of the algebraic group. This is especially true for projective varieties that have extremal projective properties: self-dual varieties, varieties of positive defect, Severi varieties, Scorza varieties, varieties of small codegree, etc. I have tried to emphasize this phenomenon in this survey. However, one aspect is totally omitted - I decided against including results about dual varieties of toric varieties and A-discriminants. This theory is presented in the beautiful book by Gelfand, Zelevinsky, and Kapranov [GKZ2] (the author always keeps it under his pillow) and I don't feel I have anything to add to it. Let us say a few words about the contents of this survey. Chapter 1 is intended as a brief introduction to projective duality. All results here were already well-known in the 19th century. After giving basic definitions in Sect. 1.1 we discuss duality for plane curves in Sect. 1.2. We give parametric equations for dual curves, discuss the connection to the Legendre transform, introduce Pliicker formulas, and study curves of degree 2, 3, and 4. In Sect. 1.3 we prove the Reflexivity Theorem. Here we follow the exposition in [GKZ2] and use the interpretation of a conormal variety as a Lagrangian subvariety in the cotangent bundle. We introduce the discriminant as a defining equation for the dual variety and the defect that measures how far the

VI

Preface

dual variety is from being a hypersurface. We prove that if the defect is positive then the variety is ruled. Finally, in Sect. 1.4 we study the behaviour of a dual variety under projections and introduce the standard notation of algebraic geometry related to divisors and line bundles. In our survey we are mostly interested in the study of varieties with symmetries, and in Chap. 2 we study projective geometry of homogeneous spaces; more precisely we look at orbit closures for algebraic groups acting on projective spaces with finitely many orbits. In Sect. 2.1 we give the necessary background on algebraic groups and fix notation. In Sect. 2.2 we discuss Pyasetskii pairing, which is an interesting instance of projective duality. We give some standard and a few exotic examples. The next Sect. 2.3 contains the systematic treatment of actions related to gradings of simple Lie algebras. These actions provide a wealth of very important varieties that will be studied throughout this book. Some examples include Severi varieties, smooth self-dual varieties, smooth building blocks for varieties of positive defect, varieties of small codegree, etc. It is quite an interesting phenomenon that varieties with extremal projective properties tend to have maximal symmetries. We finish this chapter with the description of Pyasetskii pairing for actions related to gradings of GLn called the Zelevinsky involution. In Chap. 3 we study projectively dual varieties using calculations in local coordinates. In Sect. 3.1 we prove a formula due to Katz that expresses the dimension of a dual variety in terms of the hessian of local equations of a variety. We use it to prove a formula of Weyman and Zelevinsky that expresses the defect of a Segre embedding of a product of two varieties. In Sect. 3.2 we introduce a gadget called the second fundamental form that incorporates these calculations. We prove some results of Griffiths and Harris about the second fundamental form. Wefinishthis section with the description of higher fundamental forms of flag varieties obtained by Landsberg. In Chap. 4 we study projective constructions related to projective duality but also having a merit of their own. We prove a theorem of Zak and Ran that the Gauss map of a smooth variety is a normalization. We introduce secant and tangent varieties in Sect. 4.2, prove the Terracini Lemma, give a method for the calculation of multisecant varieties of homogeneous spaces, discuss the relationship discovered by Zak between the degree of a dual variety and the order of the variety, and give an overview of old and new results related to the Waring problem for forms. In Sect. 4.3 we discuss theorems of Zak related to the Hartshorne conjecture - theorems on tangencies, on linear normality and on Severi varieties. The main tool is a connectedness theorem of Fulton and Hansen. We finish in Sect. 4.4 by explaining the Cayley trick for Chow forms. The dual variety is the image of the conormal variety which is the projectivized conormal bundle if the variety is smooth. In Chap. 5 we exploit this relation between duality and vector bundles. In Sect. 5.1 we prove a theorem of Holme and Ein about the defect of a smooth effective very ample divisor. We deduce this result from a theorem of Munoz about the dimension of the linear span of a tangential variety. We discuss the related notion of projec-

Preface

VII

tive extendability. In Sect. 5.2 we apply Hartshorne's ample vector bundles to prove a theorem of Ein that a dual variety of a smooth complete intersection is a hypersurface. We introduce resultants and prove a classical theorem that they are well-defined. We also explain the Cayley trick for resultants. In the last Sect. 5.3 we describe the "Cayley method" developed by Gelfand, Kapranov, and Zelevinsky. The idea is to show that the dual variety is represented in the derived category by Koszul complexes of jet bundles. The discriminant is then equal to the "Cayley determinant" of a generically exact complex. As an application we deduce some classical formulas for discriminants and their degrees. In Chap. 6 we discuss about the degree of dual varieties and resultants. We start in Sect. 6.1 by recalling Chern classes and then proving a formula of Katz, Kleiman, and Holme that expresses the degree of a dual variety in terms of Chern classes of the cotangent bundle. We give many examples and generalizations. We prove a theorem of De Concini and Weyman about the formula for the degree with non-negative coefficients. In Sect. 6.2 we discuss formulas for the codegree and ranks related to the Cayley method, such as a formula due to Lascoux. In Chap. 7 we study varieties with positive defect. In Sect. 7.1 we focus on beautiful theorems of Ein about the normal bundle of a generic contact locus. Since this locus is a projective subspace, it is possible to use the machinery of vector bundles on projective spaces. We prove Ein's theorem that this normal bundle is symmetric and uniform, which explains among other things a parity theorem of Landman. We introduce the Beilinson spectral sequence and use it to calculate the normal bundle to a generic contact locus in small dimensions. Finally, we study dual varieties of scrolls and prove a theorem of Ein that a variety of defect at least 2 is a scroll if and only if the normal bundle to a generic contact locus splits. In Sect. 7.2 we follow [IL] and discuss linear systems of quadrics of constant rank and how they are related to dual varieties via the second fundamental form. In Sect. 7.3 we prove a theorem of Beltrametti, Fania, and Sommese that relates the defect of a projective variety and its Mori-theoretic characteristic called the nef value. We give a brief survey of necessary results from Mori theory. We finish by giving a classification of smooth varieties of positive defect up to dimension 10 obtained by many authors and initiated by Ein. Finally, in Sect. 7.4 we use this connection with Mori theory to classify all flag varieties with positive defect. This approach was developed by Snow in contrast with the original proof of Knop and Menzel that used the Katz dimension formula. In Chap. 8 we study dual varieties and discriminants of several special homogeneous spaces. We start in Sect. 8.1 by showing how to use standard results of representation theory such as the Borel-Weil-Bott theorem, the BGG homomorphism, identities with Schur functors, and formulas of the Schubert calculus to find the codegree of Grassmannians or full and partial flag varieties. We give a list of formulas for the degree of hyperdeterminants and sketch the proof of a theorem of Zak about varieties of codegree 3. In Sect. 8.2 we

VIII

Preface

generalize the theorem of Matsumura and Monsky about automorphisms of smooth hypersurfaces to automorphisms of smooth very ample divisors on flag varieties. In Sect. 8.3 we study commutative algebras without identities from the "discriminantaP point of view. As a corollary we prove that the algebra of diagonal matrices does not have quasiderivations. In Sect. 8.4 we study anticommutative algebras (nets of skew-symmetric forms). We show that they have beautiful geometric properties related to cubic surfaces, Del Pezzo surfaces, representation theory of S5, etc. In Sect. 8.5 we show that the discriminant in a simple Lie algebra defined by analogy with the disciminant of a linear operator is equal to the discriminant of the minimal orbit, the socalled adjoint variety. Finally, in Sect. 8.6 we study related questions about schemes of zeros of irreducible homogeneous vector bundles. In particular, we address a question of classifying irreducible homogeneous vector bundles with a trivial line subbundle, find the maximal dimension of an isotropic subspace of a generic symmetric or skew-symmetric form, and study properties of the related Moore-Penrose involution. In Chap. 9 we study self-dual varieties, i.e. varieties isomorphic to their projectively dual variety. In Sect. 9.1 we consider smooth self-dual varieties. The complete list of these varieties is (conjecturally) surprisingly short. All known varieties are flag varieties so we start by considering this case, where everything follows from the classification of flag varieties of positive defect. After a brief introduction to the Hartshorne conjecture we sketch the proof of the amazing theorem of Ein that gives the complete list of self-dual varieties in the range that is allowed by the Hartshorne conjecture. We also prove a finiteness theorem of Murioz that uses the distribution of primes to give restrictions on the Beilinson spectral sequence. We finish in Sect. 9.2 by describing results of Popov about self-dual nilpotent orbits. In the final Chap. 10 we study how the topology of the variety is reflected in singularities of the dual variety. We start in Sect. 10.1 by proving the class formula and its variant due to Landman that relates the degree of the dual variety and the Euler characteristic of the variety and its hyperplane sections. In the singular case this formula was proved by Ernstrom, but the Euler characteristic has to be substituted by the degree of the Chern-Mather class. In Sect. 10.2 we prove theorems of Dimca, Nemethi, Aluffi and others that multiplicities of the dual variety are given by Milnor numbers (or classes). To give an example we follow Aluffi and Cukierman and calculate multiplicities of the dual variety to a smooth surface. Finally, we give some results of Weyman and Zelevinsky about singularities of hyperdeterminants. I would like to thank F. Zak for very inspiring discussions of projective geometry and R. Munoz for his insights about positive defect varieties. I am grateful to S. Keel, V. Popov, A. Kuznetsov, P. Katsylo, D. Saltman, E. Vinberg for many helpful remarks and encouragement. P. Aluffi, F. Cukierman, N.C. Leung, A.J. Sommese and many others sent me lots of comments about the preliminary version of this survey.

Preface

IX

Parts of this book were written during my visits to the Erwin Schrodinger Institute in Vienna, Mathematical Institute in Basel, and University of Glasgow. I am grateful to my hosts for the warm hospitality.

Moscow-Edinburgh-Austin, August 2004

Jenia Tevelev

Contents

1

Introduction to Projective Duality 1.1 Projectively Dual Varieties 1.2 Dual Plane Curves 1.2.1 Parametric Equations 1.2.2 Legendre Transformation 1.2.3 Pliicker Formulas 1.2.4 Curves of Small Degree 1.3 Reflexivity Theorem 1.3.1 Proof of the Relexivity Theorem 1.3.2 Defect and Discriminant 1.4 Projections and Linear Normality 1.4.1 Projections 1.4.2 Degenerate Varieties 1.4.3 Linear Normality

1 1 2 2 3 4 5 6 6 9 11 11 12 13

2

Actions with Finitely Many Orbits 2.1 Algebraic Groups 2.2 Pyasetskii Pairing and Kashin Examples 2.3 Actions Related to Gradings 2.3.1 Construction 2.3.2 Short Gradings 2.3.3 Multisegment Duality

17 17 32 35 35 41 49

3

Local Calculations 3.1 Calculations in Coordinates 3.1.1 Katz Dimension Formula 3.1.2 Defect of a Product 3.2 Fundamental Forms 3.2.1 Second Fundamental Form 3.2.2 Higher Fundamental Forms 3.2.3 Fundamental Forms of Flag Varieties

57 57 57 60 64 64 67 70

XII

Contents

4

Projective Constructions 4.1 Gauss Map 4.2 Tangents and Secants 4.2.1 Terracini Lemma 4.2.2 Multisecant Varieties of Homogeneous Spaces 4.2.3 degX* and ordX 4.2.4 Waring Problem for Forms 4.3 Zak Theorems 4.3.1 Theorem on Tangencies 4.3.2 Theorem on Linear Normality 4.3.3 Theorem on Severi Varieties 4.3.4 Connectedness Theorem of Fulton and Hansen 4.4 Chow Forms

5

Vector Bundles Methods 5.1 Dual Varieties of Smooth Divisors 5.1.1 Linear Envelope of a Tangential Variety 5.1.2 Dual Varieties of Smooth Divisors 5.1.3 Projective Extendability 5.2 Ample Vector Bundles 5.2.1 Definitions 5.2.2 Dual Varieties of Smooth Complete Intersections 5.2.3 Resultants 5.3 Cayley Method 5.3.1 Jet Bundles and Koszul Complexes 5.3.2 Cayley Determinants of Exact Complexes 5.3.3 Discriminant Complexes 5.3.4 Cayley Method for Resultants

89 89 89 91 93 94 94 95 96 97 97 99 102 105

6

Degree of the Dual Variety 6.1 Katz-Kleiman-Holme Formula 6.1.1 Chern Classes 6.1.2 Top Chern Class of the Jet Bundle 6.1.3 Formulas with Positive Coefficients 6.1.4 Degree of the Resultant 6.2 Formulas Related to the Cayley Method 6.2.1 Degree of the Discrminant 6.2.2 Lascoux Formula

109 109 109 110 113 114 115 115 116

7

Varieties with Positive Defect 7.1 Normal Bundle of the Contact Locus 7.1.1 Ein Theorems 7.1.2 Monotonicity Theorem 7.1.3 Beilinson Spectral Sequence 7.1.4 Planes in the Contact Locus

119 119 119 124 124 126

73 73 74 74 75 77 79 80 80 82 83 84 87

Contents

XIII

7.1.5 Scrolls 7.2 Linear Systems of Quadrics of Constant Rank 7.3 Defect and Nef Value 7.3.1 Some Results from Mori Theory 7.3.2 The Nef Value and the Defect 7.3.3 Varieties with Small Dual Varieties 7.4 Flag Varieties with Positive Defect 7.4.1 Nef Cone of a Flag Variety 7.4.2 Nef Values of Flag Varieties 7.4.3 Flag Varieties of Positive Defect

129 130 136 136 140 145 147 147 149 150

8

Dual Varieties of Homogeneous Spaces 8.1 Calculations of degX* 8.1.1 Borel-Weyl-Bott Theorem 8.1.2 Representation Theory of GLn 8.1.3 Dual Variety of the Grassmannian 8.1.4 Codegree of G/B 8.1.5 A Closed Formula 8.1.6 Degree of Hyperdeterminants 8.1.7 Varieties of Small Codegree 8.2 Matsumura-Monsky Theorem 8.3 Discriminants of Commutative Algebras 8.3.1 Commutative Algebras Without Identities 8.3.2 Quasiderivations 8.4 Discriminants of Anticommutative Algebras 8.4.1 Generic Anticommutative Algebras 8.4.2 Regular Algebras 8.4.3 Regular 4-dimensional Anticommutative Algebras 8.4.4 Dodecahedral Section 8.5 Adjoint Varieties 8.6 Homogeneous Vector Bundles 8.6.1 Zeros of Generic Global Sections 8.6.2 Isotropic Subspaces of Forms 8.6.3 Moore-Penrose Inverse and Applications

155 155 155 157 158 159 161 166 167 169 171 171 172 174 174 179 181 183 186 189 189 192 196

9

Self-dual Varieties 9.1 Smooth Self-dual Varieties 9.1.1 Self-dual Flag Varieties 9.1.2 Hartshorne Conjecture 9.1.3 Ein's Theorem 9.1.4 Finiteness Theorem 9.2 Self-Dual Nilpotent Orbits

207 207 207 209 210 212 213

XIV

Contents

10 Singularities of Dual Varieties 10.1 Class Formula 10.2 Singularities of X* 10.2.1 Milnor Numbers 10.2.2 Milnor Class 10.2.3 Dual Variety of a Surface 10.2.4 Singularities of Hyperdeterminants

219 219 222 222 224 228 231

References

233

Index

245

Introduction to Projective Duality

This chapter is intended as a brief introduction to projective duality. All results here were already well-known in the 19th century. After giving basic definitions in Sect. 1.1 we discuss duality for plane curves in Sect. 1.2. We give parametric equations for dual curves, discuss the connection to the Legendre transform, introduce Pliicker formulas, and study curves of degree 2, 3, and 4. In Sect. 1.3 we prove the Reflexivity Theorem. Here we follow the exposition in [GKZ2] and use the interpretation of a conormal variety as a Lagrangian subvariety in the cotangent bundle. We introduce the discriminant as a defining equation for the dual variety and the defect that measures how far the dual variety is from being a hypersurface. We prove that if the defect is positive then the variety is ruled. Finally, in Sect. 1.4 we study the behaviour of a dual variety under projections and introduce the standard notation of algebraic geometry related to divisors and line bundles.

1.1 Projectively Dual Varieties For any finite-dimensional complex vector space V we denote by P(V) its then lPN = projectivization (the set of 1-dimensional subspaces). If V = cNfl P(CN") is the standard complex projective space. Let VV be the dual vector space of linear forms on V. Points of the dual projective space P(V)' = P(VV) correspond to hyperplanes in P(V). Conversely, to any point p of P(V), we can associate a hyperplane in P(V)V, namely the set of all hyperplanes in P(V) passing through p. Therefore P(V) is naturally identified with P(V) . More generally, there exists the projective duality between projective subspaces in P(V) and P(V)': for any L c P(V), its projectively dual subspace L* c P(V)' parametrizes all hyperplanes that contain L. Quite remarkably, this projective duality extends to the involutive correspondence between arbitrary subvarieties in PN and (PN)' .

'"

2

1 Introduction to Projective Duality

Let X c PN be an irreducible projective variety with dimX = n. We the tangent space at any smooth point x E X,,. We can also denote by Tx,x c lPN as follows: define the embedded projective tangent space TXjx

where Cone(X) c V is the cone over X, v is any non-zero point on the line x, as a linear subspace of V (it does not depend on and we consider Tv,co,,(x) the choice of v).

Definition 1.1 A hyperplane H c PN is a tangent hyperplane of X if c H for some x E X,,. The closure of the set of all tangent hyperplanes is called the projectively dual variety X* c (lPN)'. We shall prove later that

Theorem 1.2 (Reflexivity Theorem) X**= X .

1.2 Dual Plane Curves The most classical example of a dual variety is the dual curve C* of a plane curve C c P2. By definition, generic points of C* are the tangents to C at smooth points. In this case the Reflexivity Theorem has a simple meaning: the tangent line TPat a smooth point p E C is the limit of secants for q E C as q -+ p. The point in P2 that corresponds to the tangent to C* c (P2)" at a non-singular point TPis the limit of the intersection points of the tangents Tpand Tqas q -+ p. Of course, this point is p.

1.2.1 Parametric Equations It is easy to write down a parametric representation of the dual curve C* using a given parametric representation of C. Let x, y, z be homogeneous coordinates on P2 and p, q, r the dual homogeneous coordinates on (P2)'. We choose an affine chart C2 = { z # 0) c P2 with affine coordinates x, y (so we set z to be 1).The dual chart (C2)' c (P2)' with coordinates p, q is obtained by setting the coordinate r of (P2)' to be -1. Then (C2)' consists of all lines in P2 that do not pass through the origin (0,O) E C2 c P2. Every such line either has an affine equation px qy = 1 or is the line "at infinity" with coordinates p = q = 0. Suppose that a local parametric equation of C has the form x = x(t), y = y(t), where t is a local parameter on C , and x(t), y (t) are analytic functions. The dual curve C* is parametrized by p = p(t), q = q(t), where p(t)x q(t)y = 1 is the tangent line to C at (x(t), y(t)). It follows that

+

+

Applying this formula twice one can prove the Reflexivity Theorem.

1.2 Dual Plane Curves

3

1.2.2 Legendre Transformation

Projective duality and the Legendre transformation of classical mechanics are closely related to each other. Let us recall this classical definition in the case of real functions in one variable. Details and generalizations can be found in PrI.

Definition 1.3 Let y = f (x) be a smooth convex real function, fl'(x) > 0. The Legendre transformation of f is a function g(p) defined as follows. Let x = x(p) be a point at which the graph y = f (x) has slope p. Then

Equivalently, x(p) is the unique point where the function F(p, x) = px - f (x) has a maximum with respect to x and gCp) = FCp,xb)). The Legendre transformation is easily seen to be involutive. To link it with projective duality, we need caustic curves. Let us express projective duality entirely in terms of the projective plane P2.A tangent line to a curve C at some point x is the line which is infinitesimally close to C near x. A point of (P2)' is a line 1 c P2.A curve in (P2)' is therefore a 1-parameter family of lines in P2.For example, a line in (P2)' is a pencil x* of all lines in P2 passing through a given point x E P2.The dual curve C* is a 1-parameter family of tangent lines to C.Now let us take an arbitrary curve C' c (P2)" (a 1-parameter family of lines in P2)and then find a geometric interpretation of the dual curve C1*c P2.Take some line 1 E C'.The condition that x* is tangent to C' at 1 means that 1 is a member of a family C' and other lines from C' near 1 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 C' is called the caustic curve of C'.This is nothing else but the projectively dual curve C1*. By the Reflexivity Theorem, any curve C c P2 is the caustic curve of the family of its tangent lines. Another consequence is that any 1-parameter family of lines in P2 is a family of tangent lines to some curve C. The caustic could be found (locally) using the Legendre transformation:

Theorem 1.4 Consider a family of lines y = px - g(p). Its caustic curve has an equation y = f (x), where f is the Legendre transformation of g. Proof. The tangent line to y = f (x) at a point x(p) with slope p is equal to

It remains to use the involutivity of the Legendre transformation.

0

4


1.2.3 Pliicker Formulas

Even if a plane curve C c iP2 is smooth, C* c ((iP2)' is almost always singular. There is a natural map c-+c*,p ~ ~ p . This map is a resolution of singularities. Moreover, even if C is not smooth, the curves C and C* are birationally equivalent, in particular they have the same geometric genus g. Indeed, consider the conormal variety

Obviously, Ic projects birationally on C. Therefore, by the Reflexivity Theorem, Ic also projects birationally on C*. So C and C* are birationally equivalent. If C is smooth then, of course, Ic = C. A line 1 tangent to C in at least two points is a singular point of C*. It is known as a multiple tangent. If 1 has exactly two tangency points on C and the intersection multiplicity at each of them is 2, then 1 is a bitangent. A bitangent corresponds to an ordinary double point (node) of C*. If I = TP intersects C at p with multiplicity rn 2 3, it is again a singular point of C*. If m = 3, and 1 is not tangent to C at any other point, then p is called an inflection point (or flex) of C. I is a cuspidal point (or cusp) of C*. Now we can introduce a class of "generic" curves with singularities, which is preserved by the projective duality. Namely, we say that 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, K, 6, 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*, K*, 6*, b*, f * be the corresponding numbers for C* (d* is also called the class of C). Then by the Reflexivity Theorem we have the following

Proposition 1.5 g = g*, K = f*, 6 = b*, b = 6*, f = K*. It turns out that there is another remarkable set of equations linking these numbers. It was discovered by Pliicker and Clebsch. The proof can be found in [GH].

Theorem 1.6

1 g = -(d* -l)(d* - 2 ) 2

b- f,

1.2 Dual Plane Curves

5

1.2.4 Curves of Small Degree

Conics Let C

c P2 be a smooth conic given by the equation

where A = (aij) is a non-degenerate symmetric 3 x 3 matrix. The tangent line to C at xo E C is given by (Axo,x) = 0. Hence the point E (P2)V corresponding to this tangent line has homogeneous coordinates AXO,which implies that (Ap1 0 then X is a ruled variety. Definition 1.16 A projective variety X is called ruled in projective subspaces of dimension r if for any x E X there exists a projective subspace L such that x E L c X and d i m L = r . Definition 1.17 For any projective variety X C PN and a hyperplane H , the singular locus of X n H is called the contact locus of X and H . Theorem 1.18 Suppose that def X = r 2 1. Then (i) X is ruled i n projective subspaces of dimension r. (ii) If X is smooth then the contact locus (XnH),i,, is a projective subspace of dimension r for any H E (X*),,. The union of these projective subspaces is dense i n X .

Proof. By the Reflexivity Theorem 1.7, (i) is equivalent to the following: if codim X = r + 1 then X * is ruled in projective subspaces of dimension r . By a standard closedness argument it is sufficient to check this property only for

1.4 Projections and Linear Normality

11

some Zariski open dense subset of X*. The condition for a hyperplane H to . a given x, all H be tangent to X a t x E X,, is that H contains T ~ JFor 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*. 0 (ii) is proved by the same argument. Example 1.19 Suppose that X c pN is a non-linear curve. Then def X = 0. Indeed, X obviously could not contain a projective subspace Pk for k > 0. Example 1.20 Let us give the minimal possible example of a smooth variety with positive defect. Let V = Matn,3 be the space of 2 x 3 matrices. Then and the pairing is given by the trace of the product. Let X c VV = P(V) and X * c P(V)' be the projectivizations of the varieties of matrices of rank less than 2. Then X and X * are smooth, projectively dual to each other, and both isomorphic to P1 x P2. Therefore defX = codimX* - 1 = 1. For any H E X*, the intersection X n H is the union of a smooth quadric surface and a plane. Their intersection, i.e. (X n H),i,,, is a line. See also 2.11.

1.4 Projections and Linear Normality 1.4.1 Projections Let P = P(V) be an N-dimensional projective space and L c P a projective subspace of dimension k, L = P(U), U c V. The quotient projective space P / L = P(V/U) has, as points, (Ic 1)-dimensional projective subspaces in P containing L. The projection with center L is the map

+

which takes any point x E P\ L to the (kf 1)-dimensional projective subspace spanned by x and L. It corresponds to the projection V -, V/U. P / L can be identified with any (N - k - 1)-dimensional projective subspace K c P not intersecting L. Then TL sends x E P \ L to the unique intersection point of K and the (k 1)-dimensional projective subspace spanned by x and L. Clearly, the dual projective space (PIL)" is canonically embedded in PVas the set of hyperplanes containing L, and so it coincides with L*, the dual variety of L.

+

Theorem 1.21 (i) Let X c P = P(V) be an irreducible subvariety not intersecting a subspace L = P(U) and such that dimX < dim P/L. Then (nL(X)* c L* n X*.

The discriminant A,,(x) (as a polynomial on (V/LV)~c VV) is a factor of the restriction of Ax to (V/U)'.

12


(ii) Suppose further that

TI,

:

X

-, ~

L ( X is ) an isomorphism. Then

( r L ( X ) *= L* n X*, and A,,(xl

= Ax

Proof. A hyperplane in P / L is just a hyperplane in P containing L. It is clear that, if a hyperplane H c P / L is tangent to r L ( X ) at some smooth point y = r ~ ( x )where , x E X is also smooth, then H as a hyperplane in P is tangent to X at x. This proves (i). Suppose now that r~ is an isomorphism. The same argument as above shows that, if X; c X * (resp. xL(X)T) c rL(X)*) is a dense subset of hyperplanes tangent to X (resp. r L ( X ) ) at some smooth point, then rL(X)T) = L* n X;, since r~ induces an isomorphism X,, -, rL(X),, It suffices to show that L* n X * = L* n X;. Here the assumption that r~ is an isomorphism is crucial, because otherwise critical points of r~ may produce extra components of L* n X*. Consider a germ of a curve (x(t), H(t)) in the conormal variety Ix such that x(t) E X,, for t # 0 and H(0) contains L. We claim that there is a germ (x(t), H1(t)) such that H1(0) = H(0) and and H1(t) contains L for any t. Since r~ is an isomorphism, it induces an isomorphism of Zariski tangent spaces. Therefore T ~ ( ~ does ) , ~ not intersect L. Let To c T ~ ~ be ~ a) limit , ~ position of embedded tangent Zariski spaces Tt = T,(t),x as t -, 0. Then Tt does not intersect L for any t. Therefore we may consider a family T,' of (dim L d i m X 1)-dimensional projective subspaces such that for any t we have L c T,' and for t # 0 the subspace Tl is tangent to X a t x(t). Namely, T,' is a projective subspace spanned by L and Tt. Since d i m X < dim P/L, we can embed T,' into a family of hyperplanes H,' with same properties.

+

+

1.4.2 Degenerate Varieties

Definition 1.22 An irreducible subvariety X if X is not contained in any hyperplane H.

c lPN is called non-degenerate

The next theorem shows that we may restrict ourselves to non-degenerate varieties while studying dual varieties and discriminants.

Theorem 1.23 Let X

c lPN be an irreducible subvariety.

(i) Assume that X is contained in a hyperplane H = lPN-I. If X*' is the dual variety of X , when we consider X as a subvariety of lPN-l, then X * is the cone over X*' with vertex p corresponding to H. (ii) Conversely, if X * is a cone with vertex p, then X is contained in the corresponding hyperplane H .

Proof. If HI # H is a tangent hyperplane of X then H n H' is a tangent hyperplane of X in PN-l. Conversely, if T is a tangent hyperplane of X in lPN-l then each hyperplane HI in IPN containing T is tangent to X . Therefore X * is the cone over X*'. This proves (i).


13

By the Reflexivity Theorem, we also have (ii). Namely, each hyperplane which is tangent to X * a t a smooth point necessarily contains p. Therefore X = X** is contained in the hyperplane corresponding to p. 0 In terms of discriminants, Theorem 1.23 can be reformulated as follows. Consider a surjection 71- : V --+ U. Then we have an embedding i : P(UV) v P(VV).Let X c P(UV). Then Ax is a polynomial function is a function on on U. If we consider X as a subvariety in P(VV)then V. By Theorem 1.23 these polynomial functions are related as follows:

In other words, Ai(x) does not depend on some of the arguments and forgetting these arguments gives Ax. 1.4.3 Linear Normality Theorems 1.21 and 1.23 show that in order to study dual varieties and discriminants it suffices 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 algebraic variety X is simply the sheaf of sections of some algebraic line bundle. For example, the structure sheaf of regular functions Ox 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 (Ut,gi), i E I, where Ui are open subsets of X covering X , and gi are rational functions on Ui such that gi/gj is regular on each intersection Ui n Uj. The functions gi are called local equations of the divisor. More precisely, a Cartier divisor is an equivalence class of such data. Two collections (Ui, gi) and (U,!, gi) 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 gi is regular on Ui, then we say that the divisor D is effective, and we write D 0. The subschemes {gi = 0) of the Ui can then be glued together into a subscheme of X , also denoted by D. Therefore, effective Cartier divisors can be identified with locally principal subschemes of X (locally given by one equation). Any non-zero rational function f E @(X) determines a principal Cartier divisor (f). Principal divisors form a subgroup of Div(X). Let Kx denote the sheaf of rational functions on X, ICx(U) = @(U).To every Cartier divisor D = (Ui,gi)iEIwe can attach a subsheaf Ox(D) c Kx. Namely, on Uiit is defined as g,lOu,. On the intersection Ui n Uj the sheaves g,' Ou, and gy1Ou, coincide, since gi/gj is invertible. Therefore these sheaves

>

14


can be pasted together into a sheaf Ox (D) c iCx . For instance, Ox (0) = Ox and O x (Dl D2) = OX(Dl) 8 OX(D2). The sheaves Ox(D) are invertible. In fact, multiplication by gi defines an isomorphism ("trivialization") O x (D) lu, P Ou, . This gives a surjective homomorphism Div(X) + Pic(X). Its kernel consists of principal divisors. A non-zero section of Ox(D) is a rational function f on X such that fgi is regular on the Ui for any i, in other words, such that the divisor (f) D is effective. If D itself is effective, the sheaf Ox(D) has a canonical section s o , which corresponds to the constant function 1. This can also be reformulated as follows. Let s E HO(X,L) be a non-trivial global section of an invertible sheaf L. After choosing some trivializations q5i : Lui 21 Oui on a covering (Ui), we obtain an effective divisor (Ui, 4i(si)), the scheme of zeros of s, which we denote by Z(s). For instance, if D is effective then Z(sD) = D. So basically any effective divisor is defined by one equation s = 0. However, s is not a function, but a section of an invertible sheaf. If st is another non-zero global section of L then the divisors Z(st) and Z(s) differ by the divisor of a rational function sls'. One also says that they are linearly equivalent. The sheaf Ox(-D), for D effective, is an ideal sheaf in OX. It defines D as a subscheme.

+

+

Example 1.24 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 homogeneous functions of degree d on P(V). More precisely, let n : V \ (0) -t P(V) be the canonical projection. If U c P(V) is a Zariski open set, then the sections of O(d) over U are, by definition, regular functions f on n-I (U) c V, which are homogeneous of degree d: f (Xu) = Ad 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, 0(-I), as a line bundle, is the tautological line bundle. Its fiber over a point of P(V) represented by a 1dimensional subspace 1 is I itself. If d < 0 then HO(P,O(d)) = 0. If d 0 then HO (P, O(d)) = symdVV,homogeneous polynomials of degree d. For any non-zero s E HO(P,O(d)), the corresponding effective divisor Z(s) is just the hypersurface defined by the polynomial s. In particular, hyperplanes in P(V) correspond to global sections of O(1).

>

Suppose now that X is an irreducible subvariety in PN. Then O(d) can be restricted on X , giving a sheaf Ox(d). In particular, we have a restriction homomorphism of global sections: , 3 H'(X, OX(1)). Vv = H O ( P ~O(1)) Clearly this map is not injective if and only if X is degenerate. So suppose now that X is non-degenerate. In general, res is not surjective either. Its image is a vector subspace W c HO(X,Ox(l)) with the following obvious property: for any x E X there exists a section s E W such that s(x) # 0. Divisors of the form Z(s), s E W, are just hyperplane sections X n H for various hyperplanes H .


15

In general, if X is a projective variety with an invertible sheaf L, then a family of divisors IWl = {Z(s) 1s E W c HO(X,L) is called a linear system of divisors. If W = HO(X,L), the linear system is called complete. It is denoted by ILI, or by ID1 if L = O(D). Every effective divisor linearly equivalent to D appears in ID1 exactly once. IWI is called base-point free if the intersection of all its divisors is empty, or, equivalently, if for any x E X there exists s E W such that s(x) # 0. As we have seen, if X c P(V) is a projective variety then the image of res : VV + HO(X,Ox (1)) defines a base-point free linear system. Conversely, any base-point free linear system IWI defines a morphism into a projective space. Namely, for any x E X we have a hyperplane r ( x ) c W consisting of all sections vanishing at x. This gives a regular map r : X + P(W*). If r is an embedding then W is identified with the image of a restriction map W -+ HO(X,Ox(l)) and Ox (1)is identified with L. We therefore need a final definition.

Definition 1.25 A linear system IWI 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 ILI is very ample. L is called ample if some tensor power LBm, 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 1.26 The follolowing statements are equivalent: (i) A projective variety X C P(V) is linearly normal. (ii) res : VV + HO(X,Ox(l)) is an isomorphism (iii) The embedding X C 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 10x(l)l gives a non-degenerate embedding of X into the projectivization of a higher-dimensional vector space HO(X,OX(^))^. The initial embedding is obtained from this one by the projection with center P(Im(res)*). Conversely, if X is a non-trivial isomorphic projection of a non-degenerate variety in a larger projective space P(U) then we have a proper embedding VV c UV.The map UV + HO(X,0, (I)), given by restricting linear functions from P(U) to X, is an injection, since x is non-degenerate. Thus the map res is not surjective, being the composition vVL) uVt HO(X,OX(I)). We see that in order to study dual varieties and discriminants we can restrict ourselves to linearly normal varieties. We adopt the following definition:

x

Definition 1.27 A pair (X, L) of a projective variety X and a very ample invertible sheaf L on it is called a polarized variety.

16


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 ILI.Therefore we may speak about dual varieties, defect, discriminants, etc. of polarized varieties.

Actions with Finitely Many Orbits

In our survey we are mostly interested in the study of varieties with symmetries, and so in this chapter we study projective geometry of homogeneous spaces; more precisely we look at orbit closures for algebraic groups acting on projective spaces with finitely many orbits. In Sect. 2.1 we give the necessary background on algebraic groups and fix notation. In Sect. 2.2 we discuss Pyasetskii pairing, which is an interesting instance of projective duality. We give some standard and a few exotic examples. The next Sect. 2.3 contains the systematic treatment of actions related to gradings of simple Lie algebras. These actions provide a wealth of very important varieties that will be studied throughout this book. Some examples include Severi varieties, smooth self-dual varieties, smooth building blocks for varieties of positive defect, varieties of small codegree, etc. It is quite an interesting phenomenon that varieties with extremal projective properties tend to have maximal symmetries. We finish this chapter with the description of Pyasetskii pairing for actions related to gradings of GL, called the Zelevinsky involution.

2.1 Algebraic Groups In this section we recall basic facts about algebraic groups and homogeneous spaces. Details can be found in [Hu, OVl].

Linear Algebraic Groups An algebraic variety G with a group structure is called an algebraic group if multiplication and inverse maps

are morphisms of algebraic varieties. A subgroup H c G is algebraic if H is an algebraic subvariety. In the same way we define algebraic homomorphisms.

18

2 Actions with Finitely Many Orbits

Images and kernels of algebraic homomorphisms are algebraic subgroups. If H c G is a normal algebraic subgroup then G/H is canonically an algebraic group. The commutant [HI,Hz] of two algebraic subgroups HI, H2 c G IS an algebraic subgroup of G, as are their intersection HI n Hz, their intersection, and the center Z(G) are algebraic subgroups of G. The basic example is GL(V), the group of invertible linear transformations of a vector space V. If an algebraic group G can be embedded into some GL(V) then G is called a linear algebraic group. G is linear if and only if G is an affine algebraic variety. The class of linear algebraic groups is closed with respect to taking subgroups, quotient groups and direct products. All algebraic groups in this book are linear.

Lie Algebras Any algebraic group G is smooth, and therefore G is irreducible if and only if it is connected. The connected component of G containing the unity element is an algebraic subgroup Go C G. It is called the connected component of G. The commutant of two connected subgroups in G is connected. The tangent space Te,Gto G at unity has a natural structure of a Lie algebra. We denote Lie algebras of algebraic groups by the corresponding gothic letters, e.g. by g. Any homomorphism f : GI + G2 of algebraic groups induces the homomorphism of Lie algebras df ,1 : gl + 02. If G = GL(V) then g = gI(V) is the Lie algebra of all linear operators of V with the standard commutator [X,Y] = X Y - YX. If G c GL(V) then g c gI(V) and the commutator in g is the restriction of the commutator in gI(V). A homomorphism of Lie algebras f : g1 + g2 is called algebraic if there exists a homomorphism of algebraic groups F : GI + G2 such that f is its differential.

Algebraic Actions An action of an algebraic group G on an algebraic variety X is called algebraic if the corresponding map

is algebraic. All our actions will be algebraic. If an algebraic group G acts on an algebraic variety X and Y algebraic subvariety, then the normalizer

cX

is an

is an algebraic subgroup of G. The normalizer of a point x E X is denoted by G, and is called the stabilizer of x. For example, if G acts on itself by

2.1 Algebraic Groups

19

conjugation and H C G is an algebraic subgroup (or just any subvariety), then the normalizer

and the centralizer

are algebraic subgroups of G.

Representations If we have a homomorphism G t GL(V) then we also say that we have a representation of G in V, or that V is a G-module, or that G acts linearly on V. In the same way we define representations of Lie algebras. If V is a G-module then V is also a g-module, where g is the Lie algebra of G. For example, any algebraic group G has a natural representation in its Lie algebra g called the adjoint representation Ad : G t GL(g). If G = GL(V) then Ad is just the matrix conjugation

If G c GL(V) then g c gI(V) and the adjoint representation is again given by the conjugation (restricted on 8). The differential of the adjoint representation is the adjoint representation of a Lie algebra g: ad : g t gI(g),

ad(x) = [x, -1.

If G acts linearly on V then G automatically acts on VV as well. This representation (or module, or action) is called dual. The group of all projective transformations of PN is an algebraic group denoted by P G L N + ~It . is a quotient of G L N + ~by its center. P G L N + ~also acts dually on (PN)V.In fact any automorphism of PN is given by a projective transformation, and therefore any action of an algebraic group G on P N is represented by projective transformations and corresponds to some homomorphism G -+ PGLN+1.

Invariants If G acts on the affine variety X then the ring of regular functions @[XIis an infinite-dimensional G-module. The action is defined by the formula ( 9 . f)(x) = f(gW1.x), for any g E G, x E X. This module is a union of finite-dimensional G-modules. The subring of Ginvariant (constant along G-orbits) functions is denoted by @[xIG.

20


Connected Solvable Groups The group B, of upper-triangular (n x n)-matrices is a typical example of a solvable algebraic group. Any action of a connected solvable algebraic group G on a projective variety has a fixed point (Bore1 Theorem). In particular, for any representation f : G --+ GL(V) there exists a basis such that all operators f (x), x E G, have an upper-triangular form, i.e. G preserves some flag of subspaces in V (Lie Theorem). Any algebraic group G has a unique maximal normal connected solvable algebraic subgroup Rad(G). It is called the radical of G.

Jordan Decomposition A linear operator A E GL(V) is called unipotent (resp. semisimple) if in some basis A is represented by an upper-triangular matrix with 1's on the diagonal (resp. is represented by a diagonal matrix). Let G be an algebraic group. An element x E G is called unipotent (resp. semisimple) if for some (and hence or any) embedding i : G r GL(V) the element i(x) is a unipotent operator (resp. a semisimple operator). Any homomorphism of algebraic groups takes semisimple (resp. unipotent) elements to semisimple (resp. unipotent) elements. Any element x E G has a unique Jordan decomposition

where x, is semisimple and xu is unipotent.

Unipotent Groups An algebraic group is called unipotent if all its elements are unipotent. Any unipotent group is connected and solvable. For example, the group U, of upper-triangular (n x n) matrices with 1's on the diagonal is unipotent. Another example is a vector space with respect to addition. Any character, i.e. a homomorphism G t C*, of a unipotent group G is trivial. It follows from the Lie Theorem that for any representation f : G -+ GL(V) there exists a basis such that all operators f (x), x E G, have an uppertriangular form with 1's on the diagonal. Equivalently, any finite-dimensional G-module contains a non-zero fixed vector. If a unipotent group G acts on the affine variety X then all orbits are closed. Indeed, let Y be a closure of a certain orbit. Then G has a dense open orbit in Y. If the complement to this orbit is not empty then its ideal of vanishing functions contains a non-trivial finite-dimensional G-module, and therefore a non-trivial invariant function f E C[ylG.This function is constant along the dense orbit, and therefore it is constant along Y. Hence it can not vanish on the complement to the dense open orbit. This is a contradiction. Any algebraic group G has a unique maximal normal unipotent subgroup R, (G). It is called the unipotent radical of G.


21

Reductive Groups An algebraic group G is called reductive if R,(G) is trivial. An algebraic group G is called semisimple if Rad(G) is trivial. Obviously, semisimple groups are reductive. G is reductive if and only if any finite-dimensional G-module V is totally reducible, i.e. splits as a direct sum @V, of irreducible submodules V,. If G is reductive then [G,GI is semisimple. If G is also connected then

where the locally direct product is a quotient of the direct product by a finite central subgroup. The corresponding decomposition of g is the direct product

For any semisimple Lie algebra g there exists a unique connected simplyconnected semisimple algebraic group G with a Lie algebra g. Any representation of g is a differential of the unique representation of G. All other connected semisimple algebraic groups with a Lie algebra g are quotients of G by finite central subgroups.

Standard Invariant Form Let f : g t gI(V) be an algebraic representation of a Lie algebra g of a connected algebraic group G. Then

is an invariant symmetric bilinear form on g, i.e.

It is called the standard invariant form. G is reductive if and only if (., .)v is non-degenerate for some (and hence for any) injective f . If G is semisimple then its center is finite and the adjoint representation Ad : G -+ GL(g) gives an embedding ad : g c+ gI(g). The image of g in gI(g) is a Lie algebra of derivations D E gI(g) such that

The corresponding standard invariant form is called the Killing form,

The Killing form is non-degenerate if and only if g is semisimple.

22


Algebraic Tori A connected algebraic group is called an algebraic torus if all its elements are semisimple. Any algebraic torus is isomorphic to @* x . . . x C*, where @" is the group of invertible elements in C. An algebraic torus T is obviously reductive. All irreducible representations of T are one-dimensional (any set of commuting semisimple operators can be taken into the diagonal form in some basis). Let X(T) be the group of characters of T , i.e. homomorphisms T -+ C*. This is a free abelian group, rank X (T) = dim T. X(T) admits the canonical embedding in tV by assigning to each character E X(T) its differential d x at unity. We identify X(T) with its image in tV and we write X(T) additively. Let

x

Then tV= t i +it:,

i.e. tg is the real form of tVand tRC t is the real form oft. The restriction of any standard invariant form of t on t~ is a positive-definite scalar product.

Root Decomposition Let G be a connected reductive group with a Lie algebra 0. Any subtorus of G which is maximal with respect to inclusion is called a maximal torus of G. All maximal tori are conjugated to each other. Let T be one of them. Its dimension r is called the rank of G, and the Lie algebra t of T is called a Cartan subalgebra. We have the root decomposition

where 0, = {x E 01 Ad(t)x = a ( t ) x for any t E T). Elements of A

c X(T) are called roots. We have dimg, = 1 for any

a!

E

A;

if a E A then - a E A; if a, k a E A then k = f1.


23

Positive and Simple Roots Choose any hyperplane in tg not passing through A. Roots from one halfspace are called positive and from another one negative. We denote the set of positive roots by A+ and the set of negative roots by A-. We have

There exists a unique set of simple roots 17 C A+,

such that any root a E A is an integral combination

>

with ni 0 for positive a and ni 5 0 for negative a. Simple roots are linearly independent.

Cartan Matrix and Dynkin Diagrams We choose any standard invariant form on g. Its restriction on ta is positivedefinite. Using it, we define the scalar product on t i . We use the notation

(Alp) = 2(A, l r ) for A, p E tk, p

+ 0.

for any a , p E A. Let 17 = { a l , . . . ,a,) be the set of simple roots. The Cartan matrix A = (ailaj)i,j=l,...,Y has the properties

Aii = 2,

Aij 5 O for i # j,

iff Aji = 0.

The Dynkin diagram is a graph with r vertices indexed by simple roots, where i-th and j-th vertices are connected by lAijAji 1 edges. If /Aij1 < lAji 1 then this edge has an arrow pointing from the j-th vertex to the i-th vertex. Let us summarize some further properties of the root decomposition.

E A and (a,@)< 0 then a + @ E A; if a , @

ifa,@,a+p~Aand(a,p)=Othena-PEA;

4

if x E g,, y E g-a, z E gp then [[x,Y], = (xly)(a, P

b

24


Simple Groups Any connected semisimple group is a locally direct product of simple algebraic groups, where a connected non-abelian algebraic group is called simple if all its normal subgroups are finite. G is simple if and only if g is simple (i.e. g # C and g has no proper ideals). A semisimple G is simple if and only if its Dynkin diagram is connected. Simple Lie algebras are parametrized by their Dynkin diagrams. There are four classical series A,, Bn, Cn, Dn and five exceptional cases E6, E7, Es, F 4 , and G2. An is isomorphic to sL+l, the Lie algebra of traceless endomorphisms The corresponding simply-connected simple group is SLn+l, the of en+'. group of all linear transformations of Cnfl with determinant equal to 1. The quotient of SLn+l by its center is PGLn+l. The Dynkin diagram is

Here we fix the numbering of simple roots that will be used afterwards. 2, is isomorphic to sozn+l and D,, n 4, is isomorphic to B,, n 5ozn, where so, is the Lie algebra of skew-symmetric operators in Cn with respect to the given quadratic form Q. The group On of orthogonal transformations has so, as its Lie algebra. This group has two connected components - transformations with determinant equal to 1 and -1. is called the special orthogonal group SO,. This group is not simply-connected. The Dynkin diagram of Bn is

>

>

-.1

2 3

. n-ln .-WO

The Dynkin diagram of Dn is

>

Cn, n 2, is isomorphic to 5Pzn, the Lie algebra of skew-symmetric operators in CZnwith respect to the given symplectic form w . The group Sp2, of symplectic transformations has 5p2, as its Lie algebra. This group is simply-connected. The Dynkin diagram of Cn is

-.1

2 3

. .-*on - l n

Finally, exceptional Dynkin diagrams are


25

Weyl Group For any p

# 0 we denote by

the reflection with respect to the hyperplane orthogonal to p. If a E A then r,(A) c A. The group W generated by reflection r,, a E A, is called the Weyl group. It is isomorphic to NG(T)/T, where the latter group acts on t by the adjoint representation (and dually on tV). We define the length function

There exists a unique element wo E W of the maximal length IA+ 1. The hyperplanes in t$ orthogonal to roots divide t i into cones called Weyl chambers. W acts transitively on the set of Weyl chambers, moreover, each Weyl chamber C is a fundamental domain for the action of W on t$, i.e. each W-orbit intersects C in one orbit. In particular, the number of Weyl chambers is equal to IWI. The choice of a Weyl chamber is equivalent to the choice of a set of positive roots. Namely, A+ defines the Weyl chamber

C+ ={A

E

t&I (&a) 2 0 for any a E A + } .

Of course, in this formula A+ can be substituted for the set of simple roots l7. If G is simple and of types A, D, E then all roots are conjugated to each other by the Weyl group. If G is simple and of types B, C, F, G then W has two orbits on A - the set of long roots and the set of short roots. In A - D - E cases we postulate that all roots are long.

Irreducible Representations Take any representation V of a connected reductive group G. Then we have the weight decomposition

v = X€X(T) @ vX,

where

VX= {v E V I t . v = X(t)v for any t

The subset X(T)v

= {A E

X(T) I V'

E T).

# 0)

is called the weight system of V. If V is irreducible then there exists a unique highest weight X E X(T)v such that X - p is an integral combination of simple roots with non-negative coefficients for any p E X(T)v. Moreover,

dirnvX= 1 and A E C'

26


Similarly, there exists a unique lowest weight A E X(T)" such that A - p is an integral combination of simple roots with non-positive coefficients for any p E X(T)V. The longest element in the Weyl group takes the highest weight to the lowest weight. For any character X E C+ n X(T) there exists a unique irreducible representation with the highest weight A. Highest weights of irreducible representations are also called dominant weights. We denote the semigroup of dominant weights by P+. If G is simple then Ad is irreducible, and we denote by y the highest root (the highest weight of the adjoint representation) in A+. The highest root is long.

Nilpotent Cone Take any homomorphism with finite kernel G -+ GL(V) and the corresponding embedding i : g c-+ gI(V). An element x E g is called semisimple (resp. nilpotent) if i(x) is a semisimple (resp. nilpotent) linear operator. This notion is independent of the choice of i. We have a version of the Jordan decomposition: x,, where x, is any element x E g can be expressed uniquely as x = x, nilpotent, x, is semisimple, and [x,, x,] = 0. Suppose that G is a connected reductive group. Nilpotent elements in g form an irreducible conical algebraic subvariety R(g) called the nilpotent cone. Nilpotent elements e E g are characterized by the following equivalent properties:

+

OEAd(G)e. Any Ad(G)-invariant polynomial on g of positive degree vanishes at e. e E [g,g] and [e, .] is a nilpotent operator in g[(g). (Jacobson-Morozov Theorem) There exists an da-triple (e, h, f ) C g, i.e. a triple of elements that satisfy the relations [e, f ] = h,

[h,el = 2e,

[h, f ] = -2 f .

The number of Ad(G)-orbits in R(g) is finite. For example, nilpotent orbits in gI(V) are indexed by Jordan normal forms of nilpotent linear operators.

Homogeneous Spaces If an algebraic group G acts transitively on an algebraic variety X then X is called a homogeneous space. If H c G is an algebraic subgroup then the coset space G/H is a smooth algebraic variety and the action of G on G I H by left multiplication is algebraic. Moreover, G/H is quasiprojective (open subset of a projective variety): there exists an action of G on PN such that H is the stabilizer of a point p E PN (Chevalley Theorem). In particular, G / H is identified with Gp.


27

Parabolic and Borel Subgroups Let G be a connected reductive group. Consider any algebraic subgroup H C G. If G / H is a projective variety then H is called a parabolic subgroup and G I H is called a flag variety. A maximal connected solvable subgroup of G is called a Borel subgroup. All Borel subgroups are conjugated to each other. For example, the group BN of upper-triangular ( N x N) matrices is a Borel subgroup in GLN and G I B is the variety of complete flags in CN (or in PN-I). A subgroup P c G is parabolic if and only if it contains some Borel subgroup. In particular, a Borel subgroup itself is parabolic. The Lie subalgebra b = t @ $ g, aEA+

of g is the Lie algebra of a Borel subgroup B P 3 B can be described as follows. Let

c G. Any parabolic subgroup

be some subset of simple roots. Let

denote the positive roots that are linear combinations of the roots in Then P is uniquely determined by its Lie algebra

17p.

For example, if ITp = l7 then P = G; if I l p = 0 then P = B. We let

ITGlp = l7 \ 17p and AZlp = A+ \ A:. P is a a maximal proper parabolic subgroup if and only if DGIPis a single simple root. In this case G / P is called the minimal flag variety. If V is an irreducible G-module with the highest weight X and v E Vx is a non-zero vector, then B normalizes the line in V spanned by vx, and therefore the orbit Ov of this line in P(V) is the closed G-orbit isomorphic to some flag variety. Ov is called the variety of the highest weight vectors. This is a unique closed G-orbit in P(V). If G / P is a minimal flag variety then Pic G / P = Z is generated by a very ample line bundle. Let us describe minimal flag varieties of classical groups. Projective Spaces and Grassmannians Let V = CN and G = SL(V). The projective space P(V) is identified with the flag variety G I P , where P is the normalizer in G of a line in V:

28


All line bundles on P(V) have a form O(d) for d E 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

The embedding P(V) c P ( S d v ) is called the Veronese embedding. It assigns to a line spanned by v E V the line in Sdvspanned by vd. The zero scheme Z ( s ) of a global section s E Sd(VV)Is 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 Grassmannian Gr(k, V) of kdimensional vector subspaces of V is the flag variety G / P , where P is the normalizer in SL(V) of a k-dimensional subspace in V:

k-dimensional subspaces in V are in 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 Grassmannian 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 C V is U itself. Therefore, S is a subbundle of the trivial bundle Gr(k, V) x V. The line bundle A%" is a very ample generator of Pic(Gr(k, V)) = Z. The corresponding embedding is the Pliicker embedding, Gr(k, V) c P(Akv). Namely, if U E Gr(k, V) and u l , . . . , uk is any basis of U, then U ++[ul A .. . A uk]. 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

is canonically isomorphic to SV@ In particular, the tangent bundle TGr(k,V) (V/S). Notice also that

Namely, any subspace U E Gr(k, V) maps to UL E Gr(n - k, VV). Isotropic Grassmannians Let V = C N be a vector space equipped with a non-degenerate symmetric scalar product Q or a non-degenerate symplectic form w (N should be even in this case). A subspace U c V is called isotropic if for any vl, va E U we


29

have Q(v1,vz) = 0 (resp. w(v1,vz) = 0). Let G = SO(V) or G = Sp(V), respectively. For any k 5 N / 2 let GrQ(k,V) (resp. Gr,(k, V)) denote the isotropic Grassmannian of k-dimensional isotropic subspaces. Then GrQ(k,V) and Gr,(k, V) are projective G-equivariant varieties. I t follows from the Witt Theorem that GrQ(k,V) and Gr,(k, V) are irreducible homogeneous varieties (hence flag varieties) in all cases except the symmetric case when N = 2k. In the latter case GrQ(k,C2" consists of two homogeneous components Gr$(k, C2" and GrQ(k, czk). These two varieties are in fact isomorphic to each other (as algebraic varieties, not as algebraic varieties with the action of G): if s is any orthogonal reflection then

~ ~isomorphic ) to GrQ(k - 1,C2"'). To see this let Moreover, ~ r $ ( k , @ is H = C2"l c Czk be any hyperplane such that the restriction of Q on H is non-degenerate. Then we have obvious maps

It is easy to see that both maps are isomorphisms. The variety ~ r $ ( k , C ~ ~ ) is called the spinor variety, denoted by Sk. The introduced varieties represent the complete list of minimal flag varieties of groups SO(V) and Sp(V). 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 Plucker embedding Gr,(k, V) c lP(nkv). However, A k v is reducible as a Sp(V)-module since it contains the submodule

of this submodule is an irreducible module and the The complement A$' Plucker embedding in fact gives the embedding of Gr,(k, V) in P(A!V). This embedding corresponds to the ample generator of the Picard group. Now consider the orthogonal case G = SO(V). Then again we have the Pliicker embedding GrQ(k, V) c JP(Ak~).Suppose first that

Then A k v is irreducible as a G-module, and this embedding corresponds to the ample generator of the Picard group. The most interesting case is the minimal embedding of a spinor variety.

30


Clifford Algebra Let V = C2m be an even-dimensional vector space with a non-degenerate symmetric scalar product Q. Let Cl(V) be the Clifford algebra of V, i.e. the quotient of the tensor algebra T*V by the ideal generated by elements

Cl(V) has the filtration

where C1(V)k is spanned by products

with 1 I k . The associated graded algebra is the exterior algebra AmV.The Clifford algebra Cl(V) is in fact a superalgebra, Cl(V) = c1° (v) @ c l l (V), where c l O ( v )(resp. c l l (V)) is a linear span of elements of the form vl. . . ..v,, vi E V, r is even (resp. r is odd). We define

This is a connected simply-connected algebraic group, called the spinor group. Given a E Cl(V), a = vl . . . . . v,, vi E V, let

This is a well-defined involution of Cl(V). The formula

gives the well-defined orthogonal action of Spin(V) on V. In fact, this gives the double covering Spin(V) --+ SO(V). Let U C V be a maximal isotropic subspace, and hence dim U = m. Take also any maximal isotropic subspace U' such that U @ U' = V. For any v E U we define an operator p(v) E End(A9U) p(v).ulA ... A u , = v A u l A

... u,.

For any v E U' we define an operator p(v) E End(AmU)


These operators are well-defined and therefore give a linear map p : V End(AeU). It is easy to check that

31 -+

for any v E V. Therefore we have a homomorphism (actually, an isomorphism) of associative algebras CI(V) -+ End(AeU). A'U is an irreducible Cl(V)-module but a reducible Spin(V)-module. However, its even part AeuU is an irreducible Spin(V)-module, called the halfspinor module S+.

Spinor varieties Now let us describe the embedding

It will correspond to the ample generator of Pic(8,). We are going to show that 8, is isomorphic to the projectivization of the orbit of 1in the half-spinor representation. , is an SO(V)-orbit of a fixed maximal isotropic subspace, By definition, 9 say U'. We claim that the normalizer of U' in Spin(V) (acting on V via the representation R) is equal to the normalizer in Spin(V) (acting on AeuU via the representation p) of the line spanned by 1 E Ae"U. This will show that 8, is naturally isomorphic to the projectivization of the orbit of 1 in the half-spinor representation. Consider the map

SpinV acts on V via the representation R and on Aeu and Aodd via the representation p. We claim that 9 is Spin(V)-equivariant. Indeed, for any g E Spin(V) we have

since gg = 1. It is easy to check that 9(U1, 1) = 0 and if for any v E U1 we have 9(v, w) = 0 then w = A 1 for some X E C. Suppose that g E Spin(V) normalizes the line spanned by 1 E Ae"U. Then for any v E U' we have

and therefore g normalizes U'. Suppose now that g normalizes U1. Then for any v E U' 0 = W%w(g-l)v), p(g)l) = *(v, p(g)l)1 and therefore g normalizes the line spanned by 1 E AeuU.

32


2.2 Pyasetskii Pairing and Kashin Examples In [Py] Pyasetskii has shown that if a connected algebraic group acts linearly on a vector space with finitely many orbits then the dual representation has the same property and, moreover, the number of orbits is the same. In this section we show how this result is related to projective duality.

Theorem 2.1 Suppose that a connected algebraic group G acts on a projective space Pn with finitely many orbits. Then the dual action G : (Pn)Vhas the same number of orbits. Let N

N

2=1

2=1

Pn = .u Oi and (Pn)' = .U O: be the orbit decompositions. Let Oo = Oh = 8. Then the bijection is defined is projectively dual to as follows: Oicorresponds to 0; if and only if

a

q.

Proof. Indeed, take any G-orbit O c (Pn)'. If D # (Pn)Vthen the projectively dual variety D* is G-invariant, irreducible, and non-empty. Therefore it is the closure of some orbit O' c Pn, because there are finitely many of them. 0 By the Reflexivity Theorem we have = p*.

a

Remark 2.2 If the number of orbits for the action G : Pn is infinite then, in general, there is no natural bijection between orbits in Pn and (Pn)*.However, it can be shown (see e.g. [Py]) that these actions have the same modality, i.e. the maximal number of parameters that the family of orbits can depend on. This can also be deduced from 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. preserved by homotheties. For example, if there are only finitely many orbits then the action is obviously conical. Non-zero orbits of the conical action are in a 1-1correspondence with G-orbits in P(V). The action is conical if and only if any v E V belongs to the tangent space of its orbit Tv,Gv.

Proposition 2.3 If the linear action G : V is conical then the dual action G : VV is conical as well. Proof. We can embed V = Cn and VV = (Cn)Vin Pn and (Pn)' as affine charts. These embeddings are G-equivariant. Points of VV correspond to hyperplanes in Pn not passing through the origin 0 E V c Pn. Suppose that O c VVis a non-conical orbit. Then the dual variety 8*c Pn intersects with Cn non-trivially. Therefore D* c Pn is the closure of a conical variety in Cn. Therefore its dual variety 8** c (Pn)' does not intersect (en)'. But this contradicts the Reflexivity Theorem. 0

2.2 Pyasetskii Pairing and Kashin Examples

33

Corollary 2.4 ([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 : VV has the same number of orbits. Let and

be the orbit decompositions. Then the bijection is defined as follows: Oi coris projectively dual t o responds to C71, if and only if

~(a)

~(q).

Remark 2.5 Recall that the Reflexivity Theorem is equivalent to the formula (1.2). Varieties Lag(Oi) have a nice interpretation. Namely, the representation of g (the Lie algebra of G) in V is given by the element of

The corresponding bilinear map p : V@VV-+gv

is called the moment map. Let

be its zero fiber. Then it is easy to see that if G acts on V with finitely many orbits Oi then L = Ui Lag(Oi) is the decomposition of L into irreducible components.

Fig. 2.1. Bz-orbits on uz and u i

Let us consider some examples of Pyasetskii pairing when the orbit decomposition is really complicated. They were found by Kashin [Kas]. Let Bn be the group of upper-triangular (n x n) matrices. We denote by u, the Lie algebra of upper-triangular ( n x n) matrices with zeros on the diagonal. Bn acts on u, by conjugation. This action has finitely many orbits if and only if n 5 5 (see also 8.6.1). Kashin has found all orbits in these cases, calculated Pyasetskii pairing and described Hasse diagrams for the poset of orbit closures. These results are summarised below. For n = 2 , 3 , 4 , 5 we give the Hasse diagram for the set

34


Fig. 2.2. B3-orbits on us and ug

Fig. 2.3. B4-orbits on us

of orbits in u,V and u,. We number orbits in such a way that an orbit 0: c u,V corresponds to an orbit Or c u, via Pyasetskii pairing. These diagrams have two interesting properties. Firstly, these graphs are networks, i.e. vertices are located on several levels and edges go from level i 1 to level i. This is easy to explain: the level of the orbit is just its dimension. The group B, is solvable, but all orbits of solvable groups are affine varieties and therefore have divisorial boundaries. The second property (discovered by Kashin) is much more mysterious. Namely, Hasse diagrams of u,V and u, have the same number of edges! I don't know if it is a coincidence. This question is interesting only for non-reductive groups because if G is reductive then Hasse diagrams of orbit closures in any representation V and in its dual VV are equal.

+

2.3 Actions Related to Gradings

35

Fig. 2.4. B4-orbits on ul

2.3 Actions Related to Gradings 2.3.1 Construction

A vast number of linear actions of reductive groups with finitely many orbits is ~rovidedby the following construction. Let L be a connected reductive group with the Lie algebra I. Let 8 be the center of I. Suppose that I is graded, i.e.

We also assume that 8 c Io. The cornmutant [ I , I] is automatically graded, and we have IO = [I, I]o $8, k = [t, [Ik for k # 0. It is easy to see that

is a derivation of [I,I ] . All derivations of a semisimple Lie algebra are inner, i.e. there exists a unique element [ E [I, I]o such that

Let

Lt = {X E L I Ad(x)[ = [) be the centralizer of 5 in L. Let G = (LS)O.Then I. is the Lie algebra of G and G acts on each graded piece Ik. It is easy to see that E is semisimple and G is reductive.

36


Fig. 2.5. B5-orbits on us


Fig. 2.6. Bs-orbits on u$

38


Theorem 2.6 ([Ri, Vi]) G acts on Ik (k # 0) with finitely many orbits. Proof. All elements in Ik are nilpotents. Since L has finitely many nilpotent orbits, it is sufficient to prove that any nilpotent L-orbit O intersects Ik in finitely many G-orbits. It is enough to check that, for any x E O n lk, the tangent space [Io, x] to the G-orbit Ad(G)x is equal to the intersection of Ik with the tangent space [I, x] to the L-orbit Ad(L)x. But this is clear:

and the theorem is proved.

0

Remark 2.7 The dual G-module (Ik)" is isomorphic to I L k . Indeed, take any standard invariant form (., .) on I. This form is invariant and non-degenerate, and therefore (Ik,Il) = 0 for k + 1 # 0

and it gives the G-invariant pairing between Ik and L k . The moment map

is given by the Lie bracket followed by the symmetric isomorphism I. corresponding to the restriction of (-,-) on Lo.

t

I,V

All possible gradings of I can be described as follows (see [Kac]). Fix the maximal torus T c L, the Cartan subalgebra t, the root decomposition I = t @ @ I,, and the set of simple roots Ii' = {al,.. . , a,). Take any function (YEA

and extend it to the function h : A + Z by linearity. h is called the height function. The corresponding grading of I is given by formula

Ik is a parabolic subalgebra It is clear from this description that &,O in I. This subalgebra is maximal if and only if h(ai) = 0 for all i except for some io. In this case we can also assume without loss of generality that h(ai,) = 1. Then the representation of G in II is irreducible, and hence there exists a unique closed G-orbit X c P(I1). In the following tables we look at all possible gradings of this form for simple Lie algebras. In the first column we give the type of L. Second column contains the unique i such that h(i) = 1. The third column is the description of X. The last two columns contain the defect of X and the degree of the discriminant. In all formulas we assume that k, n 2 1. There are several possible cases. By IPN we mean the projective space embedded into itself.

2.3

L

Ck+n+l if k f 2n-1

X

i

+

Actions Related to Gradings

plc

def

d

39

1

p2n-1

Table 2.1.

If XI C IPN1 and X2 c pN2,then let X1 x X2 be the product of XI 1)(N2+1)-1. This is the composition of and X2 in its Segre embedding in embeddings Xl x X2 c pN1 X pN2 C p(N1+1)(N2+1)-1 where the last embedding corresponds to the line bundle

40


I I

L

i

Dn+3 if nDn+3 is odd

or n + 3 or n + 3

Es

1

if n is even

def

X

nf2

n f 2

Gr(1,

2

Gr(1,

0

d

O

X2

I

1

4

O

Table 2.2.

In more prosaic terms, if (2; : . . . : z&,) are homogeneous coordinates on IPN1 and (2; : . . . : z%2)are homogeneous coordinates on IPN2, then the embedding PN1 x PN2 c P(Nl+1)(N2+1)-1is given by formula

((z: : . . . : ~ h , (2: ),

:

.

: z 2N 2 ) )-t

( a .

: Z:Z;

:

. . .)ij.

A smooth quadric hypersurface in Pn+l is denoted by Qn. We denote by v k ( p N ) the k-th Veronese embedding of PN in

IdN:')-'. We denote by

X I the unique closed E6-orbit in the projectivization of the 27-dimensional irreducible E6-module and we denote by X z the unique closed E7-orbit in the projectivization of the 56-dimensional irreducible E7-module.


41

2.3.2 Short Gradings

Let L be a simple algebraic group with Lie algebra I.

Definition 2.8 A grading of I with only three non-zero parts

is called short. As in 2.3, there exists a unique semisimple element J E to such that 1, = {x E I 1 [J, x] = kx). We denote by G the connected component of the centralizer of J in L. By Theorem 2.6, G acts on I*l with finitely many orbits. We are going to describe these G-orbits and the Pyasetskii pairing explicitly. These results were rediscovered several times, either using case-by-case considerations or in the invariant setting. We follow [MRS] and [Pan]. Let T be the maximal torus in G and hence in L. Then we have the partition A=A-lUAoUAl, where Ak ={a E A1 I , E Ik).

G is reductive and A. is its root system. We f kpositive roots A$ c Ao. Then

is the set of positive roots of A. Let l 7 be the set of simple roots in A+, then no= 17 n A. is the set of simple roots in A:. Let h : A + {-1,0,1) be the height function. Consider the highest root

Since y obviously belongs to Al and all n, > 0 (we shall write them down explicitly later) we see that there is a unique simple root P(Al) in Al and np(d,)= 1. In particular, the representation of G in 11 is irreducible. Moreover, if L is a simple group with the set of roots A, simple roots 17, the highest root y,and for some simple root P we have

then we can define a short grading of L with ,B = P(Al) using the height function h(P) = 1, h(a) = 0 for a E II \ {PI.

42


Now we can describe all short gradings. It suffices to draw all Dynkin diagrams marking the vertex corresponding to the simple root ,B by the number n p . If n p = 1 then this simple root defines the short grading as above. In this case we mark this vertex by a black circle.

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

is the A ~root system of the graded reductive subalgebra Then A = A - ~ u Z ~ U

Proposition 2.9 Suppose that n^l # 8. Then y E n^l and there exists a unique decomposition of gr~ded~reductive subalgebras = I' $ m such that m = m ~ 'I , is simple, and I';tl = I*l. W e denote by A' the set of roots of I'.

7

We postpone the proof until the end of this section. This proposition allows us to define the chain of shortly graded simple Lie algebras

where I("') = (I(")'. In a similar way we define A("),etc. We also define the canonical string of elements in Al:

The process terminates when A?+ l ) is empty.

Theorem 2.10 For any a E A t , let e, E g, be any non-zero vector and e-, E g-, be a vector such that (e,, e-,) = 1. W e set

and so eo

= fo = 0.

Let Ok = G . er, c Il and 0; = G fr,

c L1. Then


(i)

11 =

43

.L oi,1-1 = z.b 0:. =O

a=O

(ii) Oi c Oj if and only if i 5 j . (iii) 0: c if and only if i 5 j. (iv) Ok corresponds to Oi-k in the Pyasetskii pairing. The proof is given at the end of this section. First let us look at the examples. The following results easily follow from Theorem 2.10 by explicit calculations.

Example 2.11 Consider the short gradings of I = sI,+,. G = { ( A ,B) E GL, x GL,

Then

I det(A) det(B) = 1).

11 can be identified with Cn x Cm. There are r = min(n, m) non-zero G-orbits Oi, i = 1,.. . ,r. Oi is the variety of matrices of rank i. The projectivization of O1 is identified with X = JPn-' x JPm-l in the Segre embedding. Therefore, the dual variety X* is equal to the projectivization of the closure Or-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 Ax is the ordinary determinant of a square matrix. Another interesting case is n = 2, m 2 2: we see that the Segre embedding of JP1 x JPk is self-dual.

Example 2.12 Consider the short grading of I = so,+z that corresponds to ,6 = a l . Then G = C* x SO, and ll = Cn with the simplest action. There are two non-zero orbits: the dense one and the self-dual quadric hypersurface Q c Cn preserved by G. Example 2.13 Consider the short grading of I = 502, = D, that corresponds to ,6 = a, or ,6 = an-1. Then G = C* x SL, acts naturally on 11 = A2Cn. There are r = [n/2] non-zero orbits Oi, i = 1,. . . ,r. Oi is the variety of skewsymmetric matrices of rank 2r. The projectivization of O1 is identified with X = Gr(2, Cn) in the Pliicker embedding. Therefore - the dual variety X * is equal to the projectivization of the closure 0,-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 Ax is the Pfaffian of a skew-symmetric matrix whose degree is equal to n/2. If n is odd then def X = c ~ d i r n O , - ~= 2. Example 2.14 Consider the short grading of I = sp2,. Then G = C* x SL, acts naturally on 11 = S2Cn. There are r = n non-zero orbits Oi, i = 1,. . . ,r. Oi is the variety of symmetric matrices of rank i. The projectivization of O1 Therefore is identified with X = JPn-' in the second Veronese embedding. the dual variety X* is equal to the projectivization of the closure Or-1, the variety of matrices of rank less than r. X * is a hypersurface and Ax is the determinant a symmetric matrix. Example 2.15 The short grading of E6 gives the following action. G = C* x Spinlo and 11 is the half-spinor representation. Except for the zero orbit and

44


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 S5. It could also be described via Cayley numbers. Let Ca be the algebra of split Cayley numbers (therefore @a= 0@ C, where O is the real division algebra of octonions). Let u ++ ;ilbe the canonical involution in Ca. Let @I6 = C a e c a have octonionic coordinates u, v. Then the spinor variety S5 c IP(C16) is defined by homogeneous equations

where the last equation is equivalent to 8 complex equations.

Example 2.16 The final example appears from the short grading of E7. Here G = @* x E6, and & = C27 can be identified with the exceptional simple Jordan algebra (the Albert algebra); see [J] and also the next Example. Then E6 is the group of norm similarities and C* acts by homotheties. There are 3 nonzero 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. Example 2.17 (Severi varieties) Some of the varieties introduced previously are known as Severi varieties. They have the following algebraic description. Let DR denote one of four division algebras over R:real numbers R, complex numbers C, quaternions W,or octonions 0.Let D = DR @a @ be its complexification, and therefore D is either C, C @ C, Matz(C), or @a (the algebra of complex Cayley numbers). All these algebras have the standard involution. Let 'H3(D) denote the 3 x 3 Hermitian matrices over D. If x E 'H3(D) then we may write

xy yx, that 'H3(D) is a Jordan algebra with respect to the operation xo y = +

2

is, for any a, b we have a o b= boa

and a o ( ( a o a ) o b) = ( a o a ) o (ao b),

see [J]. In case D = Ca this is the Albert algebra . The group of automorphisms G = Aut('H3) is a semisimple algebraic group of type SO3, SL3, Sp,, or F4, respectively. The action of G on ?is 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, there exists a cubic form det on 'H3(D) defined by the formula 1 det (x) = - ((Tr x ) ~ 2 73(x3) - 3(Tr x) 'Tr(x2)) 6

+


45

Consider the subgroup in SL('FI3) preserving det. Its connected component is called the group of norm similarities G. It is a semisimple algebraic group of type SL3, SL3 X SL3, SL6, or E6,respectively. The representation of G on 'F13 is irreducible. The projectivization of the cone of the highest weight vectors is called the Severi variety corresponding to ID. The Severi variety corresponding to @a is the minimal flag variety of E6 embedded into P26 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 IDa = R then 'F13 is equal to the Jordan algebra of symmetric 3 x 3 complex matrices and det is the standard determinant. G = SL3 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 P2 in the second Veronese embedding. If DR = C then 'F13 is equal to the Jordan algebra of 3 x 3 complex matrices and det is the standard determinant. G = SL3 x SL3 acts on matrices by left and right multiplication, and the Severi variety is the projectivization of the cone of rank 1 matrices. Therefore this Severi variety is isomorphic to P2 x P2 in the Segre embedding. If IDR = W then 'F13 is equal to the Jordan algebra A2C6 with the multiplication given by a1oa;, = a1 A a z , where we identify A2C6 and A4C6 by means ). of a standard symplectic form w (it is easy to see that A4C6 = w A A ~ C ~The function det is defined as det(a) = a A a A a (in the matrix form det is equal to the Pfaffian of the skew-symmetric matrix corresponding to a ) . The Severi variety is the projectivization of the variety of decomposable bivectors of the form u A v, u, v E C6. Therefore this Severi variety is isomorphic to Gr(2, C6) in the Plucker embedding. For IDw equal to R, @, or W it is possible to define higher dimensional Severi varieties in the same way (isomorphic to Pn in the second Veronese embedding, Pn x Pn in the Segre embedding, or Gr(2, C2n+2)in the Plucker embedding). However, for IDR = 0 only the case of 3 x 3 matrices makes sense. The last theorem of this section gives a geometric interpretation of the inductive process based on Proposition 2.9:

Theorem 2.18 Consider the minimal nontrivial orbit X = P(O1) c P(I1) and let z E X . Then the quotient projective space P ( I ~ ) / Tis~ isomorphic ,~ to P(1i) and the projection with center TX,xtakes P(Ok) to P(Oi) for k < r . The proof is given below.

Proof of Proposition 2.9. Suppose that (@(Al),6) = 0 for some

Since y is the highest root, 0 I m, 5 n, for any a E no. Then

46


0 for any a,,B E All K1 = m @ I:, where

m=

r;

r,

@

=

@

L.

We take x = ep,. Since Dl is long, if a E Al then (a,PI) > 0 if and only if a - ,B is a root. Therefore m is the cone over T%,x.It follows that the quotient projective space ~ ( l ~ ) / is ? equal ~ , ~ to P(Ki). By Theorem 2.10 and Lemma 2.22, y = ep, eg, . . . ep,,, E Ok.

+ + +

Since y E I;, we see that .rr(P(Ok))> P(Ok), where .rr is the projection with center T,J. Suppose that this inclusion is strict. Since the image of .ir is G'-invariant, it follows that

i.e. there exists m E m such that m lim exp(th) . m

t+-03

Therefore z E

+ z E Ok. Let h = [ep,, e-p,]

= 0,

E

t. Then

lim exp(th) . z = z.

t+-oo

c, but this contradicts Theorem 2.10.

0

2.3.3 Multisegment Duality Apart from actions associated with Zgraded semisimple Lie algebras, another class of actions with finitely many orbits is provided by the theory of representations of quivers (see [Gal). These two classes overlap: the representations of quivers of type A give the same class of actions as the standard gradings of GL(V). The Pyasetskii pairing in this case was studied under the name of the

50


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, of pairs of integers (i, j) such that 1 5 i 5 j 5 r. Let Z: denote the semigroup of families m = (mij)(i,j)Es of non-negative integers indexed by S. We regard a pair (i,j) E S as a segment [i,j] = {i, i 1 , . . . ,j) in Z. A family m = mij E Z s can be regarded as a collection of segments, containing mij copies of each [i,j]. Thus, elements of Z: can be called multisegments. The weight Iml of a multisegment m is defined as a sequence y = {dl,. . . ,d,) E ZT, given by

+

di=

x

mkl

for i = 1,.. . ,r.

In other words, Iml records how many segments of m contain any given number i E [l,r].For any y E Z1; we set Z;(y) = {m E Z: I Iml = y). Another important interpretation of Zg(y) is that it parametrizes isomorphism classes of representations of quivers of type A. Let A, be the quiver equal to the Dynkin diagram of type A,, where all edges are oriented from left to right. Let A: be the dual quiver with all orientations reversed. A representation of A, with the dimension vector y = {dl, . . . ,d,) E ZT+ is a collection of vector spaces Cdl, . . . ,Cd7 and linear maps

A representation of A: with the dimension vector y = {dl,. . . ,d,) E ZI; is a collection of vector spaces Cdl, . . . , cdrand linear maps

Therefore representations of A, with dimension vector y are parametrized by points of a vector space V(Y>= Representations of A; with dimension vector y are parametrized by points of a vector space r-1

~ ( y ) "= @ ~ o m ( @ ~ ' cdi). +l, i=l

Notice that the vector spaces V(y) and V(y)" are naturally dual to each other as vector spaces and as modules of the group

with respect to the natural action. These actions could also be described in terms of graded Lie algebras. Namely, consider the Lie algebra g = gIdl+...+d, .


51

Elements of 0 can be represented as block matrices with diagonal blocks of shapes dl x dl, . . ., d, x d,. This determines Zgrading by diagonals: block diagonal matrices have grade 0, etc. Then go is a Lie algebra of G(y) and the action of G(y) on V(y) and V(y)" is equivalent to the action of G(y) on 0-1 and 01. Orbits of G(y) on V(y) (resp. V ( Y ) ~correspond ) to isoclasses of representations of A, (resp. A): with the dimension vector y. These orbits are parametrized by elements of Z$ (7). Elements of S parametrize indecomposable A,-modules (or A:-modules). Namely, each (i, j) E S corresponds to an indecomposable module Rij with dimension vector

and all maps are isomorphisms, if that is possible, or zero maps otherwise. Then any family (mij) E Z; corresponds to an A,-module (or A:-module) es~?. B; the Pyasetskii Theorem 2.4, there is a natural bijection of G(y)-orbits in V(y) and V ( T ) ~Therefore . there exists a natural involution C of ~ $ ( y ) , which can be extended to a weight-preserving involution of 25: called the multisegment duality. The inductive description of was given in [MW]. Let m = (mij) be a multisegment of weight y = (dl,. . . ,d,). We set il = min{i I di # 0) and define the sequence of indices jl , . . . ,jp as follows:

j t , mil+t,j # 0),

where t = 1,.. . , p - 1. The sequence terminates when jp+1 does not exist. Let it = il t - 1 for t = 1,.. . , p 1. We associate to m the multisegment m' given by

+

+

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

< i 5 j 5 r.

Theorem 2.23 ( [ M w ) If the multisegment m' i s associated to m then

The involution C 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]. An explicit description of was found in [KZ]. To formulate it, we need the following definition. For any multisegment m E Z: the ranks rij(m) are given by

dim U - dim(@)"

= cl - k2.

This argument shows that

It remains to find a form such that this inequality becomes an equality. We shall proceed by induction. Suppose first that cl and c2 are positive. Take a form

such that corank(cp1) = max(0, cl

-

k2,c2 - kl).

We can define the form cp on

such that its restriction on

@

@

coincides with cp',

ekz-l)I(e, f ) ,

Then it is clear that cp E Bil(cl, kl; c2, k2) and

Now we may assume that cl = 0. Suppose that c2 and kl are both positive. Take a form cp' E Bil(0, k1 - 1;cz - 1,k2 - 1)

3.1 Calculations in Coordinates

63

such that corank(cpl) = max(0, c2 - k1). We define the form cp on

such that the restriction of cp on Ckl-I

@ Ckzpl coincides with

cpl ,

and d e , e) = cp(e, f ) = 11 d f , f ) = 0. It is clear then that cp E Bil(0, kl; ca, k2) and

It remains to consider only the cases when either kl = 0 or cl = ca = 0. In both cases lemma is obvious. [7 It follows immediately from Theorem 3.5 that the same result holds for the product with any number of factors. Let Xk c PNk, k = 1. . . ,r, be irreducible projective varieties. Then X = X1 x . . . x X, is embedded in JPN1 x . . . x PNrand the latter is embedded in JPN via the Segre embedding, where N 1 = (Nl 1) x . . . x (N, 1). This gives an embedding X c JPN. Let X* c (JPN)" be the dual variety.

+

+

+

Corollary 3.7

In particular, X* is a hypersurface if and only if

Example 3.8 The Segre embedding

identifies cone(IPL1x . . . x PLr)with the variety of decomposable tensors in CLifl @ . . . @ CLr+l.By Theorem 3.5, (JP1l x . . . x JP1-)* is a hypersurface if and only if 1, for k = l , ... , r . 21kIl1+

...+

The corresponding discriminant is called a hyperdeterminant for the matrix format (Il 1) x . . . x (1, 1).

+

+

64

3 Local Calculations

3.2 Fundamental Forms 3.2.1 Second Fundamental Form

To measure how T , , ~ moves to the first order, one calculates

It is easy to see that (x) c Ker(dy),(v) for any v E T,,x, and therefore d(y), factors to a map

Example 3.9 Let us give a useful coordinate description of IFF2. It will show that PF2 was implicitly used in the proof of Theorem 3.1. For x E X , we choose a homogeneous coordinate system To,. . . ,TN for PN SO that x = (1 : 0 : . . . : 0) and T,,X is spanned by the first n+ 1 coordinate vectors. Let XI = Tl/Tol . . ., x~ = TN/To be the corresponding affine coordinate system for the affine chart of PN. Then X I , . . . , x, is a holomorphic coordinate system for X centered at x, dxl,. . . ,dx, is a basis of T l x , and d / d ~ , + ~. .,. ,d/dxN is a basis of NX,p(V) 1,. For p = n 1, . . . ,N we have in the neighborhood of x that xll lx vanishes to the second order at x. Therefore

+

xPlx =

C qijuxixj + (higher order terms).

i,j=l Then

FlFz is symmetric, essentially because the Gauss map is already the derivative of a map and mixed partial derivatives commute.

Definition 3.10 The section FF2 of the vector bundle S2T: @ Nx,p(v) constructed above is called the projective second fundamental form of X . If X c P(V) is singular then IFF2 is defined over X,,. It is convenient to consider IFF2 as a map PF2 : N$,p(V) t S2Tz and to set IFF2/ = P(FF2(N&V))). One can think of IFF2/, as a linear system of quadric hypersurfaces in P(T,,x). The space P(N;,p(V) ) ,1 has the geometric interpretation as the space of hyperplanes tangent to X at x, i.e., the hyperplanes H such that X n H is singular at x. Then FIF: can be viewed as a map that sends a hyperplane to the quadratic part of the singularity of X n H at x. Let Base IFF21, c P(TXlx)be the base locus of a linear system IFF2/,, i.e. the variety of directions tangent to lines osculating to order 2 at x (that is, X appears to contain these lines to the second order). So we have

3.2 Fundamental Forms

65

Let Sing IIFP21, be the set of tangent directions such that the embedded tangent space does not move to the first order in these directions. Thus

Clearly this is just the kernel of the differential of the Gauss map. And so by Theorem 4.2, if X is smooth, then Sing IIFIF21, = 0 for any x. A weaker result that Sing IIFP21, = 0 for generic x E X was proved in [GH]. However, these results do not imply that generic quadrics from the second fundamental form are not degenerate. In fact, the following theorem is just a reformulation of Theorem 3.1: Theorem 3.11 If X is any projective variety and x E X,,, then the projective second fundamental form IPP21xof X at x is a system of quadrics of rank bounded above by dim X - def X . Moreover, if x E X,, is a generic point and Q c IFF21, is a generic quadric, then rank Q = dim X - def X . A useful formalism for dealing with second (and higher) fundamental forms is provided by moving frames. We recall it briefly; more details and applications can be found in [GH]. In PN a frame is denoted by {Ao, . . . ,AN). It is given by a basis Ao, . . . ,AN for c N + l . The set of all frames forms an algebraic variety 3 ( P N ) . The general linear group G L N + ~acts on 3 ( P N ) transitively with trivial stabilizer (it is . of the vectors Ai may be the principal homogeneous space of G L N + ~ )Each viewed as a mapping v : 3 ( P N ) + CN+l. Expressing the exterior derivative dv in terms of the basis {Ai) gives

+

The (N 1)2differential forms wij are called the Maurer-Cartan forms (on the group G L N + ~ )Taking . the exterior derivative of this formula we obtain the Maurer-Cartan equations

Geometrically we may think of a frame { A o , .. . , AN) as defining a coordinate simplex in PN, and then wij gives the rotation matrix when this coordinate simplex is infinitesimally displaced. There is a fibering 7r : 3 ( P N ) + PN given by

66


1-forms wi = woi are horizontal for T, i.e. they vanish on fibers of think of T as defined by the foliation

T.

We may

The equation N

dAo =

C

N

w i ~= i

C

W ~ Amod ~

A.

has the following geometric interpretation. For each choice of the frame {Ao,.. . ,AN) lying over p E IPN, the horizontal 1-forms wl, . . . ,WN give a basis for the cotangent space TLpN.The corresponding basis vl, . . . , VN E Tp,pv for the tangent space has the property that vi is tangent to the line AoAi. Assume now that we are given a connected n-dimensional smooth variety M c IPN, not necessarily closed. Associated to M is the submanifold 3 ( M ) c 3(BN) of Darboux frames

defined by the conditions that A0 lies over some p E M and Ao,A l l . . . , A, span Tp,M, the embedded tangent space of M at p. On F ( M ) we have the condition that differential 1-forms w l , . . . ,wn give a basis of T;M and wi = 0 for i > n. Therefore it is natural to think of the wl, . . . ,wn as homogeneous coordinates in the projectivized tangent spaces P(TPlM).For example, a quadric in IP(Tp,~)is defined by an equation Cyj=l qijwiwj = 0 with qij = qji. Since wi = 0 for i > n, we also have dwi = 0 on 3 ( M ) . On the other hand, w j A wji. Therefore, for any by the structure equations we have dwi = i > n, we have

c:,

71

Since wl, . . . ,wn are linearly independent, an easy calculation (called the Cartan Lemma) shows that, for any p > n and any i = l , .. . , n, we have

Then for any p

=n

+ 1,.. . ,N we set

An easy local calculation (see [GHl]) shows:

Proposition 3.12 The linear system of quadrics in IP(Tp,~)spanned by Q,, p = n 1,. . . ,N, is the projective second fundamental form IFF2I.

+


67

If we think of IFIF2 as the map S 2 T p ,4~ ( N M , P ~and ) p identify ( N M , P ~ ) p with CN+l/cone(TP,M).Then for any vector v E T p ,the ~ vector IFIF2(v)is given in coordinates by

3.2.2 Higher Fundamental Forms

Definition 3.13 If { p ( t ) ) is any holomorphic arc in IPN described by a vector-valued function Ao(t), then the osculating sequence is the sequence of linear spaces spanned by the following collections of vectors:

We then have the Euler-Meusnier Theorem

Theorem 3.14 ([GHl]) For v E T p , ~the ? vector IFF2(v,v ) E ( N M , P ~ ) p gives the projection in Cn+'/ one(^^,^) 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 n

3 = dt

(2)

A, mod Ao. i=l Recall that wi is a 1-form on the frame manifold. Hence the expression 2 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 {wl, . . . ,wn) as a basis in the cotangent space T I M then this equation shows that 2 is equal to the i-th coordinate of v E T p ,in~ the dual basis {vl , . . . , vn) of Tp,M(in fact, vi = Ai mod Ao). Differentiating further, we get d2Ao

;iii- =

2 (2)(2)

mod Ao,. . . ,An

i=l

c c (2)(2) n

=

N

qijp i,j=l @=n+l

But we also have

mod c o n e ( % ~ ) .

68


This finishes the proof. 0 The higher fundamental forms are defined similar to the second fundamental form IFF2. The most geometric way is to utilize the Euler-Meusnier Theorem 3.14 as the definition. Suppose that M c PN = P(V) is a smooth variety, p E M , and TP,M c V is the cone over T p , ~Then . the sequence of osculating spaces

$: is the linear span of k-th is defined as follows. The osculating space T osculating spaces for smooth curves p(t) c M , p(0) = p. The k-th normal is defined as v/T~$,so we have = N M Ch. ~ Suppose that space N J * ~

NJ~L

~ T ~ ~ & is/ any T ~ tangent ~ ~ vector and {p(t)} is any holomorphic v E T p ,= arc in PN with tangent vector v at t = 0. Let A(t) c V be a curve projected to p(t). Then it is easy to see that the vector

depends only on v. This construction gives a map

called the k-th projective fundamental form. Of course, it can also be described via moving frames, in coordinates, by differentiating the Gauss map, etc., see ( k ) can, e.g. [GHl, Lan41. The sequence of dimensions of osculating spaces Tp,M of course, depend on p E M. One solution is to restrict to an open subset of M where these dimensions are fixed.

Example 3.15 ([GHl]) Suppose that Pm x Pn c IP'nm+n+m is the Segre embedding. Recall that if Pm = P(V) and Pn = P(W), then the image M of the Segre embedding is the image of the natural inclusion P(V) x P(W) c P(V €3 W). Choosing coordinates {X,) in V and {Y,) in W, this inclusion is given by the mapping ({X,), {Y,)) -+ {X,Y,). We denote by {Ao,.. . ,Am) and {Bo,. . . ,B,) frames for P(V) and P(W). We shall use the range of indices 1 5 i, j 5 m, 1 5 a,,D 5 n. If A. lies over vo E P(V) and Bo over wo E P(W), then the frame

in P(V €3 W) lies over vo €3 WO. If we write dAo =

C OiAi mod Ao, i

then

dBo =

C $,B, a

mod Bo,


69

Therefore (3.4) is a Darboux frame for M and

Differentiating further, we get

Therefore the second fundamental form is given by

~ ~ 2 ( v o @ w + v @ w o , v o @ w + v @ w o ) = 2 v @mod w V@wo+vo@W In terms of homogeneous coordinates IFF2I is the linear system of quadrics

{+i)

for TvO,pmand

{+,I

for T,,,p,

for any matrix qi,. In other words, matrices of quadrics from this linear system have the form

where Q is an arbitrary matrix. Summarizing, we see that the projectivization of the tangent space at any point x E M is Pn+m-l and IFF21, is the complete linear system of quadrics having as a base locus the union Pm-I U Pn-I of two skew linear subspaces.

Example 3.16 ([GHl]) Suppose that Pn = P(V) c P ( S d v ) is the d-th Veronese embedding. In terms of homogeneous coordinates {Xo, . . . , Xn) for P(V), the embedding is given by

where FAvaries over a basis for the homogeneous forms of degree d. Given a point vo E Pn, e.g., vo = (1 : . . . : 0), vo maps to (1 : . . . : 0); the projectivization of the tangent space at vo is Pn-l spanned by first n coordinate vectors in IP(sdv). The affine coordinate system in the affine chart containing vo is given by F~(x)/x$ = Fx(X1/Xo,. . . ,Xn/Xo). Therefore IPF21,, is the linear system of all quadrics.


70

3.2.3 Fundamental Forms of Flag Varieties

Let G be a connected semisimple complex algebraic group, g its Lie algebra, and Ug the universal enveloping algebra of g, i.e. the quotient of the tensor algebra T'g by the ideal generated by elements

Ug has a natural filtration Ug(0) c Ug(l) c . . . such that Ug(" is spanned by products X I . . . xl, where xi E g and 1 k. The associated graded algebra is the symmetric algebra Sym* g of g. There exists a G-equivariant linear isomorphism n : symk g -+ U g ( " / ~ g ( " ~ ) given by

(~ec"' X)* and by the Reflexivity Theorem the following filtration descends strictly: i

4.2.2 Multisecant Varieties of Homogeneous Spaces

Example 4.7 Let V = C"' @ C1+', where k 2 1. Let X c P(V) be the projectivization of the variety of rank 1 matrices. Then X is isomorphic to the Segre embedding of Pk x P1. It is easy to see that seen-' X is the projectivization of the variety of matrices of rank at most n. Indeed, if A and B are two matrices and rkA = n, rk B = 1 then rk(A B) n 1, and for generic A and B we have

+ < +

The degree of Secn X is given by Giambelli's formula [Full:

Since ord(X) = min{k

+ 1,1+ I),

if k = I then ord(X) is equal to deg X*, i.e. to the degree of the determinant. If k > 1 then X* is not a hypersurface and deg X * =

(7').

If an algebraic group G acts on PN with finitely many orbits, then for any two orbits 01 and 0 2 the join S(01,02) is G-invariant and therefore is equal

76

4 Projective Constructions

to for some orbit 0 . This gives an interesting associative product on the set of G-orbits. For example, the following theorem generalizes the previous example. Let L be a simple algebraic group and P a parabolic subgroup with an abelian unipotent radical. In this case I = Lie L admits a short grading with only three non-zero parts: I = L1$ I. $ 11. Here lo $ I1 = Lie P and exp(h) is the abelian unipotent radical of P. Let G c L be a reductive subgroup with Lie algebra 6.By Theorem 2.10 G has finitely many orbits in II naturally labeled by integers from the segment [0,r]. Let Xi = P(Oi), i = 0 , . . . , r.

Theorem 4.8 S e ? X 1 = Xk+l. Proof. We use the notation of Theorem 2.10 and argue by induction on r and k. By induction, Seckx1= S(X1, Xk). Moreover, Seckx1is obviously a union of G-orbits, and hence ~ e c " ~ > Xk+1. Suppose that this inclusion is strict, i.e. SeckXI > Xk+a Let k 2 < r. By Theorem 2.18, if x E X1 then the quotient projective space P ( I ~ ) / T is~ isomorphic ,~~ to P(Ii) and the projection with center 'l;,,xl takes Xi to X,! = P(0;) for i < r. Therefore ~ec"[ > XL+2, but this contradicts the induction hypothesis. Let k 2 = r. In this case our assumption means that

+

+

+

By the Terracini Lemma 4.5, if x E 01 and y E Or-2 are generic then Tx,o, Ty,O,-z = 11. By Theorem 2.18, the quotient projective space P ( I ~ ) / T is~ , ~ ~ isomorphic to P(Ii) and the projection with center ?z,xl takes Xr-2 to Xi-,. 0 It follows that T ~ ( , ) , ~ ; - ~= P(I;), and this is the contradiction.

Example 4.9 Consider the Pliicker embedding

Then SeckX is the projectivization of the variety of skew-symmetric matrices of rank at most 2k 2. Therefore

+

ord X =

I:[

.

If n is even then ordX = deg X* = n/2 is the degree of the Pfaffian. If n is odd then codimX* = 3 and

(see [HTl]).

4.2 Tangents and Secants

77

Example 4.10 Consider the second Veronese embedding

~ec" is the projectivization of the variety of symmetric matrices of rank at most k 1. Therefore ord(X) = n = deg X*.

+

Example 4.11 Consider the spinor variety

Then Sec X = PI5.

Example 4.12 Let C27be the exceptional simple Jordan algebra, and let

be the projectivization of the variety of elements of rank one. Then SecX is the cubic hypersurface (defined by the norm in the Jordan algebra) and sec2X = P26.Notice that ord(X) = 3 is again equal to deg X*.

4.2.3 degX* and ordX The previous section gives many examples of varieties with deg X * = ordX. Here we study the relation between these invariants more closely.

Definition 4.13 Let Z c PN be a projective variety, and let z E Z. We denote by mult, Z the multiplicity of Z at z. If L is a general (codimZ)dimensional subspace of iPN passing through z, then L meets Z at z and deg Z - mult, Z other points. Furthermore, if U is a small neighborhood of z in Z and L' is a general (codim 2)-dimensional subspace of JPN sufficiently close to L, then L' meets U in mult, Z nonsingular points. We denote by sing" those points of Z with multiplicity at least k 1.

+

Multiplicity gives the stratification

Theorem 4.14 ([Z7])( ~ e c 'X)* C sing"*.

Proof. We argue by induction. Assume for simplicity that X * is a hypersurface, the general case being similar. We will show that, for a generic point Hk E (~ec"' X)* and its arbitrary neighbourhood U in X*, a general line close to HI, intersects U in at least k points. Let (xl, . . . , xk) be a general collection of Ic points of X , let


78

and let HI, E P,, n . . . n ~ , , .

Then Hk E (seek-I X)* by the Terracini Lemma. It suffices to show that multHkX * t.By induction, for a general point

>

>

in a neighborhood of Hk we know that multH,-, X * k - 1. Now we join HkP1to a general point H E Pxkclose to Hk.This line meets X * in finitely many points, since otherwise X * is the cone over Hk-l and X is degenerate. Perturbing this line a little we see that

Theorem 4.15 ([Z7])

>

(i) degX codimX+ 1. (ii) degX* 2 o r d X + d e f X .

Proof. (i) We argue by induction on codimX. Let x E X be a smooth point and let 7r : JPN --+ JPN-l be the projection with center x. Then degX = deg7r(X)

+ 1 2 codimn(X) + 2 = codimX + 1.

(ii) We are going to show that degX*

> k+defX+

1,

(4.1)

where k is the minimal integer such that s i n g k x * = 0. Then the claim will follow from Theorem 4.14. If def X = 0, then there is nothing to prove. Suppose that def X > 0, and let H E singk-' X * be a general point. Let 7r : ( P ~ --+ ) ~ be the projection with center at H , and let X*' = 7r(X*). Since X is nondegenerate, X * is not a cone, and so dim X*' = dim X * and degx*' = degX* - multH X * = degX* - k. Moreover,

x*'= cod imp^)^ X *

C O ~ ~ ( ~ N - I ) V

>

+

-

1 = def X,

and so deg X*' def X 1 by (i). Thus we have (4.1). 0 It is conjectured that the inequality in this Theorem becomes an equality if and only if X is a variety of rank 1 elements in a simple Jordan algebra.

4.2 Tangents and Secants

79

4.2.4 Waring Problem for Forms We finish this section with an overview of several results about multisecant varieties of Veronese embeddings. Let

be the image of the d-fold Veronese embedding, where N = (n;d) - 1. Multisecant varieties of Xd," were studied since the middle of the 19-th century most notably due to the following observation. If 1 1 , . . . , 1, are linear forms then the sum of powers f = l f + ...+ 1; belongs to the cone over sets-' Xd,n and the general point of sets-' Xd," has this form. For instance, if we want to find the order of Xd,", we have to solve the following problem: to find the minimal s such that the general form of degree d in n 1 variables can be decomposed as a sum of s powers of linear forms.

+

Theorem 4.16 ([AH]) ordXd,, =

I&

except

IT];

d = 2, where ordXd,, = n + 1 instead of d = 3 and n = 4, where ordXd,, = 8 instead of 7; d = 4 and n = 2, where ordXd,, = 6 instead of 5; d = 4 and n = 3, where OrdXd,, = 10 instead of 9; d = 4 and n = 4, where ordXd,, = 15 instead of 14; All exceptions were classically known [C, Rel, Re2, Re3, Ric, Pa, Dix]. is an integer and s = ordXd,,. Then a Suppose now that s = generic form of degree d in n 1 variables has finitely many presentations as a sum of s powers of linear forms. In several classical cases this presentation is unique:

&("id) +

0

n = 1, d = 2k - 1, s = k, see [Syll]. n = 2, d = 5, s = '7, see [H, Ric, Pa]. n = 3, d = 3 , s = 5 , see [Syll, Rel, Re2, Re3].

It is not known if there are any further examples. Let us explain these results in the most simple case of binary forms, i.e. when n = 1. The reader may wish to look a t [RaS] for further results and references.

Theorem 4.17 Let f be a binary form of degree d in variables x, y. Then where linear forms aix + biy are distinct and non-trivial, if and only if


80

Proof. Since ( b i g - ai$)

. (six

+ b i ~ =) 0,~

the first part of the theorem is clear. Let Si denote the vector space of binary forms of degree i, where we set Sf = 0 for i < 0. Then D(Sd) c Sd+ and it is easy to see that actually D(Sd) = Sd-s (for instance, use induction on s). Therefore dim Ker D 1 sd = st, where s' = min{d 1,s). Since the forms

+

(a12

+ bl Y ) ~.,. . , (aslx + bs/ y)d

are linearly independent (consider the Vandermond determinant), they span and everything is proved. 0 Ker D Let S,I denote the vector space of differential operators of degree i with constant coefficients. Then any f E Sdinduces a linear map

,sI

L -+ Sd-s, D Af : S

H

D .f.

Example 4.18 Let d = 2k, s = k+ 1. Then dim S L = k+2 and dim Sd-s= k. It can be verified that dim KerAf = 2 for generic f , and generic differential operators in Ker Af have no multiple roots. Therefore general forms of degree 2k have P1presentations as a sum of k 1 powers of linear forms.

+

Example 4.19 Let d = 2k, s = k. Then dims: = k+1 and dimSd-s = k+1. Then det Af = 0 is an irreducible hypersurface, and its generic points are exactly forms that admit the presentation as a sum of k powers of linear forms. Therefore seck-l X2,t4 is a hypersurface of degree k 1.

+

Example 4.20 Let d = 2k+l, s = k+l. Then dims: = k+2 and dimSd-s = k 1. It can be verified that dim Ker A = 1 for generic f , and Ker A is spanned by a differential operator without multiple roots. Therefore general forms of degree 2k + 1 have a unique presentation as a sum of k 1 powers of linear forms.

+

+

+

Example 4.21 Let d = 2k 1, s = k. Then dims: = k k 2. It can be verified that

+

+ 1 and dimSd-s =

{fIrkAf < k + 1 )

+

+

is an irreducible subvariety of codimension 2 and of degree (k 2)(k 1)/2; cf. Example 4.7. Its generic points are exactly forms that admit the presentation as a sum of k powers of linear forms. Therefore ~ec"' X2k+l,l has codimension 2 and degree (k 2)(k 1)/2.

+

+

4.3 Zak Theorems 4.3.1 Theorem on Tangencies Definition 4.22 Suppose that X C PN is a smooth n-dimensional projective variety. The tangential variety TanX is the union of all (embedded) tangent spaces to X

4.3 Zak Theorems

81

Both TanX and SecX are closed irreducible subvarieties of PN, with Tan X c Sec X , and

Theorem 4.23 ([FH], [Z2]) Suppose that X variety. Then either

c PN is a smooth projective

or Tan X = Sec X. This theorem has the following generalization. Suppose that Y c X c P N are arbitrary irreducible projective varieties. The relative tangential variety Tan(Y,X) is the union of tangent stars T;X for y E Y, where the tangent star T,*X for x E X is the union of limit positions of secants (XI,xu), where xt,x" E X and xl,x" -+ x.

Theorem 4.24 ([Z2])Suppose that Y varieties. Then either dim Tan(Y, X ) = dim X

+ dim Y,

cX

C P N are irreducible projective

dim S(Y, X) = dim X

+ dim Y + 1

The following theorem is called the Zak Theorem on tangencies in the literature.

Theorem 4.25 ([Z2],[FL]) (i) Suppose that X c PN is a smooth nondegenerate projective variety with dimX = n. If L is a k-plane i n PN and k 2 n then

(ii) IfX c IFN is a non-linear smooth projective variety then dim X * 2 dim X . If X * is smooth, then dimX = dimX*.

Proof. (i) Indeed, let Y = (L n X),i,g. Then x E Y if and only if T ~ c J L. Therefore Tan(Y, X ) c L. In particular, dim Tan(Y, X) < k . On the other hand, since X is non-degenerate, X Ff L. Therefore S(Y, X ) 6 L. In particular, Tan(Y,X) # S(Y,X). By Theorem 4.24 it follows that dimTan(Y, X ) = dim Y n. It follows that dim Y < k - n. (ii) Let H c (X*),,. Then ( X n H),ing is the projective subspace of dimension def X . Therefore, by (i) we have

+

def X

< dim H - dimX = codimX - 1.

82


It follows that dimX I :dimX*. If X * is also smooth then, by the Reflexivity Theorem and the same argument as above, dimX* 5 dimX, and therefore 0 dim X = dim X *. Next we prove that if X is smooth then X is a normalization of its Gauss image y (X) : Proof of Theorem 4.2. Indeed, it follows from Theorem 4.25 that y, is finite, i.e. any n-dimensional projective subspace L is tangent to X at finitely many points. On the other hand, Theorem 1.18 implies that a generic tangent 0 space L is tangent to X along a projective subspace. The following theorem generalizes Theorem 4.25 (i) to the singular case.

Theorem 4.26 ([Z2]) Suppose that X c IPN is a nondegenerate projective variety with Gauss images yk (X) . If L c yk (X) then

where we set dim XSing = -1 i f X is smooth. 4.3.2 Theorem on Linear Normality Theorem 4.27 ([Z4]) Suppose that X projective variety with dim X = n. If

then Sec X

c PN is a smooth non-degenerate

= PN.

+

Proof. By Theorem 4.23, either TanX = SecX or dimSecX = 2n 1. Suppose that Sec X # IPN and dim Sec X = 2n 1. Then 2n 1 < N, which contradicts codim X < Suppose now that TanX = SecX and SecX # P N . Let z E TanX be a generic point and Q = {x E X I z E TX,x}.

y.

+

+

Then Q is closed and an easy dimension count shows that dim Q = 2n - dim Tan X. For any point x E Q we have

z

E ?x,x

c TanX

and therefore

TZ,XC T z , ~ a n X . It follows that Tan(Q, X ) c T ~ , ~ , , However, (~). S(Q, X) $ T ~because , T z , T a n X is a proper subspace and X is not degenerate. By Theorem 4.24 we have

~

4.3 Zak Theorems

Since S(Q, X)

83

c SecX, we have dimS(Q,X) 5 dimSecX. Then

y.

+

0 Hence 2 N > 3 n 1. But this contradicts codimX < Theorem 4.27 is called the theorem on linear normality due to the following corollary first conjectured in [Ha2].

Corollary 4.28 Suppose that X variety. If

c PN is a smooth non-degenerate projective

N+2 codimX < 3 then X is linearly normal.

Proof. Recall from Sect. 1.4 that if X is not linearly normal then there exists a subvariety X' c PN+l and a point p E PN+l such that X is isomorphic to X' via the linear projection with center p. In particular, SecX' # PNf l. But this contradicts Theorem 4.27. 17 4.3.3 Theorem on Severi Varieties Zak [Z2] has also classified the varieties in the borderline case codimX

=

-.N + 4 3

Theorem 4.29 ([Z4]) Suppose that X c PN is a smooth non-degenerate ndimensional projective variety such that codim X = Then Sec(X) = PN except for the 4 following cases:

y.

0 0

0 0

n = 2, X = P2, X C P5 is the Veronese embedding. n = 4 , X = P2 x P2, X c P8 is the Segre embedding. n = 8 , X = Gr(2, C6), X c PI4 is the Plucker embedding. is the projectivization of the highest weight vector orbit n = 16, X c i n the 27-dimensional irreducible representation of E6.

The varieties listed in Theorem 4.29 are called Severi varieties after Severi who proved that the unique Zdimensional Severi variety is the Veronese surface. Scorza and Fujita-Roberts [FR] have shown that the unique 4dimensional Severi variety is the Segre embedding of P2 x P2. It is also shown in [FR] that n r 0 mod 16 when n > 8. Tango [Tan] proved that if n > 16 then either n = 2a or n = 3 . 2b, where a 7 and b 5. The 16-dimensional Severi variety was constructed by Lazarsfeld [La]. All Severi varieties (1)-(4) arise from Example 4.8. Another description of Severi varieties can be found in Example 2.17.

>

>

84


4.3.4 Connectedness Theorem of Fulton and Hansen The rest part of this section is occupied by the proof of Theorem 4.24. We are going to use the Connectedness Theorem of Fulton and Hansen:

Theorem 4.30 ([FH]) Let f : X -t BN be any morphism, where X is irreducible and projective. Then f -'(Lo) is connected for any projective subspace LOc PN such that dim f (X) > codimp Lo. Proof. We may assume that f is finite. Indeed, for a non-finite f , let f = go h be the Stein factorization (see e.g. [Hal]), i.e. h : X -+ XI has connected fibers, and g : XI + PN is finite. Then dimX1 = dim f (X) and if g-l(Lo) is connected then f (Lo) is also connected. Let k = dim Lo. Let U c Gr(k,PN) be the open set of subspaces L such that ~ o d i m ~f (X) ( ~ )n L = codimp L, and let Uo c Gr(k, BN) be the set of all subspaces L such that f -l(L) is irreducible. Uo is non-empty by Bertini's theorem (see [Hal] or [Wei]).It follows from Zariski's principle of degeneration (see [Full or [Za]) that f-'(L) is connected, non-empty, of codimension dimX k - N for all L E U. For each L E U, there is a positive cycle f *(L) on X whose support is f-'(L). This determines a morphism from U to the Chow variety Z of cycles on X . Let

-'

+

be the closure of the graph of this morphism, n : r + Gr(k, PN) the projection. Since Z is projective, .~ris proper as well as birational. Zariski's principle of degeneration implies that for any y E r the corresponding cycle C, is connected (i.e. its support IC,I is connected). Zariski's main theorem implies that n-'(Lo) is connected. It follows that the union of IC,I for y E r-l(Lo) is connected. It remains to prove that

The inclusion > follows from the continuity of limit cycles. So it remains to prove the inclusion c.Let x E f -l(Lo) and let z = f (x). Consider the subvariety G of Gr(k,BN) consisting of all subspaces passing through z (therefore G is isomorphic to Gr(k - l,IPN-I)). An easy dimension count shows that G meets U. Let c r be an irreducible subvariety that maps birationally to G. For each L c G n U, the cycle f *(L) contains x, and therefore for any y E r1the cycle C, contains x. In particular, there exists the limit cycle Cy for y E r1n n-I (Lo) that contains x. 0 Now we derive several corollaries.

Theorem 4.31 ([FH]) Let A C PN x PN be the diagonal. If X is an irreducible variety, f : X + PN x PN a morphism, and dim f (X) > N then f ( A )is connected.

4.3 Zak Theorems

85

Proof. Suppose that f is a finite morphism, a general case is handled using the Stein factorization. Assume that fP'(A) is not connected and choose points x and 2' in its different connected components. Choose a hyperplane H in PN not containing any of the factors of f (x) or f (x'). Let Po be the open set of PN x PN consisting of those points none of whose factors lie in H . Then Po = @2N = P2N\ {zO= O), where zo, . . . , Z ~ are N the homogeneous coordinates on P2N. Let W c IPN x IPN x P2N be the closure of the graph of this birational correspondence. If coordinates are chosen on PN such that H is the hyperplane xo = 0, and the homogeneous coordinates on the k-th copy of PN are x:, . . . ,x k , then a point (xl) x (x2)x (z) E PN x PN x lF'2N belongs to W if and only if there are constants XI, X2 such that

Hence W is defined by the equations

Let a : W -+ PN x IPN and p : W -+ be the two birational projections. Let L be the N-dimensional linear subspace of P2N defined by the equations

Then it is easy to check that Let x be the irreducible component of the fiber product X maps onto X. Then there is a diagram

where the square commutes, and that

IT

X P N ~ ~W N

that

is surjective. Then equations (4.2) imply

f -'(A) 3 ~(g-'p-'(L)), and f -'(A

n PO) c .ir(g-ljT1 (L)).

(4.3)

Now dim,@(x) = dim f (X) since f (X) meets the open set Po which corresponds isomorphically to an open set in Then by Theorem 4.30 we know that (Pg)-l (L) is connected. Therefore ~ ( ( P g ) - (L)) l is connected. But by (4.3) r((pg)-l (L)) is then a connected subset of f (A) which contains both 0 x and x'. This is a contradiction.

-'

86


Corollary 4.32 ([FH]) (i) Let Y be an irreducible subvariety of the projective space PN and let f : X -+ PN be a morphism from an irreducible variety X to PN with dim f (X) > cod imp^ Y. Then f (Y) is connected. (ii) If X and Y are irreducible subvarieties of PN with dimX + dimY > N then X n Y is connected. Proof. (i) We apply Theorem 4.31 for the map F : X x Y + PN x PN given by F = (f, Id). 0 (ii) Follows from (i). Corollary 4.33 ([FH]) (i) Let X be an irreducible variety of dimension n, and f : X -+ PN an unramified morphism, with N < 2n. Then f is a closed embedding. (ii) Let X be an irreducible n-dimensional subvariety of PN with N < 2n. Then the algebraic fundamental group of X is trivial (i.e. X has no nontrivial itale coverings). Proof. (i) If f : X --+ PN is unramified, consider the product mapping f x f : X x X --+ PN x PN.The unramified assumption is equivalent to the assertion that the diagonal Ax in X x X is open as well as closed in (f x f ) - l ( A p ~ ) .Since f is automatically finite, dim(f x f ) ( X x X ) = 2n, ) connected by Theorem 4.31. Hence and so (f x f ) - l ( A p ~ is

implying that f is one-to-one and is therefore a closed embedding. (ii) Follows from (i).

+

Proof of Theorem 4.24. If dim Tan(Y, X ) = dim Y dim X then the Theorem is obvious because S(X, Y) is irreducible. So assume that dim Tan(Y, X ) < dim Y + dim X. Assume also that Tan(Y, X )

# S(X, Y).

Let L be a generic subspace of PN such that dim L

+ dim Tan(Y, X ) = N - 1,

L n Tan(Y, X ) = 0.

But L meets S(X, Y) at a point

Let n : PN -+ PN'be the projection with center L, where N' = dim Tan(Y, X). Let X' = n(X), Y' = n(Y). Since X n L = 0, nlx is finite and therefore

4.4 Chow Forms

+

87

+

dim X' dim Y' = dim X dim Y > N'. Hence by Theorem 4.31, Y x x , X is connected. We claim that Y x x , X is supported on Y x Y. This will be ~ . Suppose that Y x x j X a contradiction because clearly (yo,xo) E Y x x X is not supported on Y x Y. Then for any (y, x) E (Y xxt X ) \ (Y x Y), the secant (x, y) intersects L. Therefore, for each point

(the set of these points is non-empty by connectedness!), the embedded tan0 gent star at y intersects L, which contradicts L n Tan(Y, X ) = 0.

4.4 Chow Forms Let X C P(V) be an irreducible subvariety and P1 = P(U). By Theorem 3.5, ( X x P1)* is a hypersurface in P(V 8 U)V for def X 5 1 5 dim X . Since we always have def X 5 dimX, this gives a system of associated hypersurfaces. We can identify P(V @J U)' with the projectivization of the space of linear maps Hom(U, VV). We abuse notation and denote a non-zero linear map and its projectivization by the same letter. Let P(V €3 U): be an open subset consisting of operators having maximal possible rank 1 1. For any f E P(V 8 U): we can associate two projective subspaces

+

and the orthogonal subspace U(f ) l c P(V). Recall that I$ c P(V) x P(V)" is an open part of the conormal variety consisting of pairs (x, H) such that x E Xsm and H is the hyperplane tangent to X at x.

Theorem 4.34 ([WZ]) ( X x P1)* is the closure of the set of points f E P(V 8 U): such that

In two extreme cases 1 = dim X or 1 = def X this result can be made slightly more precise. Consider the set Z ( X ) of all ( N - n - 1)-dimensional projective subspaces in PN that intersect X . This is a subvariety in the Grassmannian Gr (N - n - 1,PN).

Theorem 4.35 Z(X) is an irreducible hypersurface. Proof. Let B(X) c X x Z ( X ) be an incidence variety consisting of pairs (x, L) such that x E X , L E Gr(N - n - l , P N ) , and x E L. Consider projections p : B(X) -, X and q : B(X) -+ Z(X). Then p is a Grassmannian fibration, and any fiber of p is isomorphic to Gr(N - n - 2, PN-I). In particular, B(X) (and hence Z(X)) is irreducible. On the other hand, q is birational, because

88


a generic ( N - n - 1)-dimensional projective subspace intersecting X meets X in exactly one point. Now an easy dimension count shows that Z ( X ) is a 0 hypersurface.

Definition 4.36 2 ( X ) is called the Chow form of X. Now we can relate Chow forms and dual varieties. We continue to use the notation of Theorem 4.34. The following theorem is called a Cayley trick.

Theorem 4.37 ([WZ]) Let n = dim X and n* = dim X * . Let 1 = dimX, and pl : P(V 18U ) ; -+ Gr(N - n - l , P N ) be the map f H ~ ( f ) - ' - .Then ( X x JP1)* = p , 1 ( 2 ( ~ ) ) . Let 1 = def X , and pa : P(V 18U): + Gr(N - n* - 1,PN) be the map f ++ U(f). Then (X x P1)* = p , l ( Z ( ~ * ) ) . We refer the reader to [GKZ2] for further results on Chow forms.

Vector Bundles Methods

The dual variety is the image of the conormal variety which is the projectivized conormal bundle if the variety is smooth. In this chapter we exploit this relation between duality and vector bundles. In Sect. 5.1 we prove a theorem of Holme and Ein about the defect of a smooth effective very ample divisor. We deduce this result from a theorem of Munoz about the dimension of the linear span of a tangential variety. We discuss the related notion of projective extendability. In Sect. 5.2 we apply Hartshorne's ample vector bundles to prove a theorem of Ein that a dual variety of a smooth complete intersection is a hypersurface. We introduce resultants and prove a classical theorem that they are well-defined. We also explain the Cayley trick for resultants. In the last Sect. 5.3 we describe the "Cayley method" developed by Gelfand, Kapranov, and Zelevinsky. The idea is to show that the dual variety is represented in the derived category by Koszul complexes of jet bundles. The discriminant is then equal to the "Cayley determinant" of a generically exact complex. As an application we deduce some classical formulas for discriminants and their degrees.

5.1 Dual Varieties of Smooth Divisors 5.1.1 Linear Envelope of a Tangential Variety

Here we study the relation of a dual variety of a projective variety to a dual variety of its hyperplane section. We start with an auxilliary result.

Definition 5.1 Let X c PN = P(V) be a smooth projective variety. Let V be y x : X -+ Gr(n,PN), n = dimX, be the Gauss map, let S c O 18 the universal bundle on Gr(n, BN), and let Gx = y*(S). The fiber of Gx over x E X is equal to the tangent space to ConeX along the generatrix corresponding to x.

90

5 Vector Bundles Methods

Theorem 5.2 ([Mun3]) Let X c IPN be a smooth projective variety and Y c X a connected subvariety of X . Then dim(Tan(Y, X)) = N - h0(N:,~~ (1) IY). Proof. The obvious exact sequences

give rise to the exact sequence

Let (Tan(Y, X)) = P(W)

c P(V). Then there is a commutative diagram

Any choice of splitting induces the splitting of Nx,p~(-l)ly as a direct sum of F and Oy 8 V/W. This implies that hO(~:,pN (1)ly) 2 h 0 ( o y @ V/W)

= N - dim(Tan(Y, X)).

Since there is an injective map

HO(N:,~,(~)~~)can be seen as a vector subspace UV c VV giving a split sequence 0 -+ Fl -+ N X , p ~ ( - l ) (+ ~ U @ Oy -+ 0. So we can build a diagram as before:

5.1 Dual Varieties of Smooth Divisors

91

It shows that (Tan(Y,X ) ) c P(Ker(V + U)) . Therefore h O ( ~ & P(1)l N y ) = dim U 5 N - dim(Tan(Y, X)). The theorem is now proved.

0

5.1.2 Dual Varieties of Smooth Divisors Suppose that X c PN is a smooth projective variety. The following theorem gives the defect of smooth hyperplane or hypersurface sections of X. Theorem 5.3 ([E2, HK, Hol, Mun31) (i) Let Y = X

nH

be a smooth hyperplane section of X . Then def Y = max(0, def X - 1).

Moreover, if X* is not a hypersurface, then Y* is a linear projection of X* with center H . (ii) If Y is a smooth divisor corresponding to a section of Ox(d) for d 2 2 then Y* is a hypersurface. Proof. (i) Let us suppose first that def X

> 0. Take a linear projection

associating to any hyperplane HI of IPN the intersection HI n H . of an HI E X * with Since def X > 0, the contact locus L1 = (X n X intersects Y. Hence HI is tangent to X at a point of Y, and so H1 n H is tangent to Y. This implies that IT(X*) c Y*. On the other hand, a general H1 E Y* is tangent to Y at y E Y. Then the , is tangent to X at z and 17(Ml) = HI. Hence hyperplane MI = ( T ~ , xHI) n ( x * ) = Y*. Let us show that the map nix* is generically one-to-one. Suppose that H' E Y* is a general tangent hyperplane, L = (X n is a contact locus, and HI, H2 E X * are such that 17(Hl) = 17(H2) = HI. Then each Hi is tangent to X at a point yi E L and

But since dim(Tan(L,Y)) = ( N - 1)=

l~(1)) =

(N - ~ O ( N $l~(1))) , ~ ~ - 1 = dim(Tan(L, X)) - 1

by Theorem 5.2, we have

92


for any point y E L. This implies that H1 = Hz. It follows that dimX* = dimY*, and hence def Y = def X - 1. Now we are going to show that a smooth hyperplane section of a zero defect variety is a zero defect variety. Suppose the opposite and take the following divisor in the conormal variety Ix:

in the linear system I prT (Ox (1))1. Since H' is tangent to X at y for any (y, HI) E IXIY, HI n H is tangent to Y at y. But k = def Y > 0 and so it is tangent to Y along a k-dimensional projective subspace L for which

Ix1

Then

is contained in the set of points

dim pr2(Ix 1 y ) = N - 2 - dim pr;l (F)

with F a generic point in the image, and therefore dim pr2(Ix 1 y ) 5 N - 2 - k. It follows that we can choose an irreducible curve C in X * such that

and nzl(C) is again a curve because, if H' E X* is such that the contact locus of X with H' has a poisitive dimension, then this contact locus must intersect Y and then HI E pr2(Ix ly ). This implies that

By the projection formula we have the contradiction

because prl p r ; l ( ~ ) is again a curve. (ii) Here we shall consider Y as a subvariety in IPN, not in H. Then contains IXly as a divisor D. Using the exact sequence

Iy

and IY= P(N;,pN), IX1 y = P(N$,PNIY), Ny,x = Oy (-d), we see that Oh (D) 21 pr; OY(1- d) @ pr; Oy. (1).

(*>

5.1 Dual Varieties of Smooth Divisors

93

Consider the morphism g given by the complete linear system I pra Oy*(l)l. Then Y* is a linear projection of g(Iy). Thus dimY* = dimg(Iy). We claim that glI,\D is one to one and, therefore, dimY* = dimg(Iy) = dim Iy = N - 1 and Y* is a hypersurface. Indeed, let y1 and y2 be two distinct points in I y \ D. We need to show that the linear system I prz Oy*(1)l separates yl and ~ 2 If. pr1(yl) = pr1(y2) then pr2(yl) # pr2(y2) because pr2 maps pr;l(prl(yl)) isomorphically to a projective subspace in Y*. So we may assume that prl(yl) # pr1(y2). Then we can find a hypersurface S of degree d - 1 in PN which contains prl(yl) but not pr1(y2). By (*), D p r , l ( ~ ) E I prz Oy*(I)/. But

+

yl E D+pr,l(S)

and y2

6D + p r l l ( S ) .

Thus yl and ya are separated by I pra Oy*(1) 1.

5.1.3 Projective Extendability Definition 5.4 A nonsingular variety X c PN is called projectively extendable if there exist a variety X' c PNfl and a hyperplane L c PNfl such that L intersects X' transversally, L n X' = X and X' is not a cone. In this case X' is called a projective extension of X . X c PN is called smoothly extendable if X' is smooth. In general X can be extended in many different ways:

Example 5.5 ([Z6]) Let X = v4(P1) C P4 be the rational normal curve of degree four. Then X is a hyperplane section of the Veronese surface v2(P2) which is projectively inextendable (see below). On the other hand, X is a hyperplane section of the rational scroll F2of degree four in P5 and of P1 x v2(P1). Both F2and P1 x v2(P1) are hyperplane sections of a scroll X C P6 with fiber P2 over P1 which in its turn is a hyperplane section of the Segre variety P1 x P3 c P7; generic sections of P1 x P3 by five-dimensional linear subspaces of P7 are projectively isomorphic to P1 x vz(P1) while special nonsingular sections are isomorphic to F 2 . It is not true in general that extendability implies smooth extendability:

Example 5.6 ([Z6]) Let X' = v2(Sz(v3(P2))), where v, is the Veronese map of order m and S, (v3(P2)) c PI0 is the cone with vertex z 6v3(P2)) over the Del Pezzo surface v3(P2) C P9. Then Kxr 2 Ox!(-1) and X' C P38 is a Fano threefold of degree 72 with a unique singular point. Let X C P37 be a generic hyperplane section of XI. Then H1(X, O x ) = 0, K x = 0 and X is a nonsingular K3-surface. If X had a nonsingular projective extension x', then X' would be a nonsingular Fano threefold of degree 72 while by Iskovskih's theorem the degree of such a threefold does not exceed 64. Therefore the K3-surface X is smoothly inextendable.

94


Recall the definition of the vector bundle G x from the proof of Theorem 5.2. It is clear that HO(X,Gx) = 0 since a nonzero section of Gx would yield a point from n x E x ? x = ~ 0. Thus the exact sequence

~ where shows that the natural homomorphism a! : V + HO(X,N x I p(-1)), V = CN+l and P(V) = iPN, is a monomorphism and

Theorem 5.7 ([Z6]) Suppose that hO(X,NX,*N(-1)) = N X is a quadric or X is inextendable.

+ 1. Then either

Sketch of the proof. We observe that, if X is a hyperplane section of a variety XI c PN+', then the condition of the theorem is satisfied for almost all hyperplane sections of X'. This means that for a general hyperplane M c the variety pr2(Ix/lxlnM) is contained in a hyperplane in (PN+')'. Since X' is not a cone, (X1)*is non-degenerate, and from the Bertini theorem applied to the preimage of the linear system 1 0 ( e ~ + ~ ) v (on l ) lIxt it follows that X is a hypersurface. On the other hand, it is clear that quadric is the 0 only hypersurface satisfying the condition of the theorem. The connection with dual varieties is also clarified by the following theorem. Theorem 5.8 ([Z6]) If X * is normal, then X is inextendable. It should be noted that normality of X * is not necessary for inextendability of X . In fact, let X = v,(Pn), where n 2 2, m 2 3. Then X * is not normal, but X is inextendable. The last assertion (due to Scorza) can be derived from the following theorem.

Theorem 5.9 ([Z6]) If H1(X, Tx(-1)) = 0, then either X is a twisted cubic curve or a quadric or X is inextendable.

5.2 Ample Vector Bundles 5.2.1 Definitions

The notion of ampleness was extended to vector bundles by Hartshorne [Ha3]. Let E be a vector bundle on X . We consider the variety XE = P(EV), the projectivization of the bundle EV.There is a projection p : XE + X whose fibers are projectivizations of fibers of EV,and a natural projection n : EV\ X -+ XE, where X is embedded into the total space of EVas the zero section. We denote by J(E) the tautological line bundle on XE defined as follows. For open U c XE, a section of E(E) over U is a regular function on s - I (U) which

5.2 Ample Vector Bundles

95

is homogeneous of degree 1 with respect to dilations of E V . The restriction of [(E) to every fiber p-l(x) = P(E,V) is the tautological line bundle 0 ( 1 ) of the projective space P(E,V).

Definition 5.10 E is called ample if [(E) is an ample line bundle on XE. If E is itself a line bundle then XE = X, [(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 also holds for vector bundles: E is ample if and only if, for every coherent sheaf F , there is an integer no > 0, such that, for every n no, the sheaf F@Symn(E) is generated by its global sections as an Ox-module . Using this description it is easy to see that the direct sum of ample vector bundles is ample, etc.

>

5.2.2 Dual Varieties of Smooth Complete Intersections

Theorem 5.11 ([El])I f X c PN is a nonlinear smooth complete intersection then X* is a hypersurface. Proof. We may assume that X is nondegenerate. Let dimX = n. Then X is scheme-theoretic intersection of N - n hypersurfaces HI, . . . , H N - ~such that deg HI, = dk > 2. Therefore

It follows that N x , p (-1) = Ox (dl - 1) @ . . . CBO x ( d ~ - , - 1) is an ample vector bundle. Let

Ix

= P(Nz,pN(I))

c PN x

(IPN)"

be the second projecbe the conormal variety and let pr : Ix + tion. Then pr* O(1) is the tautological line bundle L on P(N$,pN ( 1 ) ) .Since N X , P(-1) ~ is ample, it follows by the standard argument that pr is a finite morphism onto its image X*. Indeed, if pr is not finite then it contracts a curve C. Therefore L . C = 0 but then nL . C = 0 for any n and nL cannot be very ample. It follows that dimX* = dimIx = N - 1 and X* is a hypersurface.

96


5.2.3 Resultants Classically, discriminants were studied together with resultants. Let X be a smooth irreducible projective variety and let E be a vector bundle on X of rank k = dimX+l. Set V = HO(X,E). We shall assume that E is very ample, i.e. ((E) is a very ample line bundle on XE. In particular, ((E) (and hence E ) is generated by global sections. Notice that HO(X,E) = H O ( X ~ , ( ( E ) ) , and therefore (XE)*. On the contrary, suppose that v E V. Then the corresponding section s E HO(X,E) vanishes at some point x E X . Therefore s defines a linear map ds : TXtx -t T X , =~ TX,x@ Ex. Let pr : T X , -+ ~ Ex be the ~) in some hyprojection. Since dimX = dim E - 1, pr o d ~ ( T ~is, contained perplane H c Ex. An easy local calculation shows that the hyperplane in P(VV)corresponding to v contains the embedded tangent space to XE at the point corresponding to H. Therefore V = (XE)*. Now let us prove that V is a hypersurface. A dimension count shows that it suffices to prove that a generic section s of HO(X,E) vanishing at x E X does not vanish at other points. In other words, a generic hyperplane in P = P(VV) containing some fiber F of Ex does not contain other fibers. Consider the linear projection nF : P \ F -t PIF. If F1is any fiber of XE distinct from F then xF(F1) is a projective subspace of dimension dimF1 = k - 1. Therefore images of fibers of XE distinct from F form an at most (k - 1)-dimensional family of projective subspaces of dimension k - 1. An easy dimension count

5.3 Cayley Method

97

shows that there exists a hyperplane H C P / F that does not contain any projective subspace from this family. The preimage of this hyperplane in P contains only one fiber F . 0

5.3 Cayley Method 5.3.1 Jet Bundles and Koszul Complexes Let X be a smooth irreducible algebraic variety with an algebraic line bundle C. We consider the bundle J(C) of first jets of sections of C. By definition, the fiber of J(C) at x E X is the quotient of the space of all sections of C near x by the subspace of sections which vanish at x with their first derivatives. In other words, J(C), = C/Z~L, where 1,is the ideal of functions vanishing at x. Thus J(C) 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 1:L. The correspondence f H j ( f ) is C-linear but, being a differential operator, it is not Ox-linear. Let us summarize without proofs some properties of jet bundles.

+

Theorem 5.15 ([GKZ2, KS]) (i) For any line bundles C and M on X , there exists a canonical isomorphism

(ii) There exists a canonical exact sequence of vector bundles

(iii) Let X = P(V). Then the J(Ox(l)) is a trivial vector bundle Ox 8 VV The relevance of jets to dual varieties is as follows. Suppose that X c P(V) is an irreducible projective variety. We take L = Ox(l). Then any f E VV is a global section of C. The following result is an immediate consequence of the definitions.

Proposition 5.16 f E VV represents a point in the dual variety X * if and only if j ( f ) vanishes at some point x E 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 Koszul complexes of sheaves on X:

98


The differential in IC+ is given by the exterior multiplication with s, and the differential in IC- is given by the contraction with s, that is, by the map Aj Ev -+ Aj-l EVdual to the map -4j-l E + AjE given by As. We fix gradings in K+ and IC- by assigning the degree j to AjE in IC+ and by assigning the degree -j to AjEV in IC-. Notice that we have

IC- (E, s)

-

IC+(E, s) 8 ArEV[r],

where [r]means the shift in the grading by r. The following well-known theorem shows that the cohomology of Koszul complexes 'represents' the vanishing set of s.

Theorem 5.17 (i) IC+(E, s) or IC- (E, s) are exact if and only if the scheme of zeros Z(s) is empty. (ii) Suppose that X is smooth and that s vanishes along a smooth subvariety Z(s) of codimension r = r a n k E and, moreover, s is transversal to the zero section. Then L ( E , s) has only one non-trivial cohomology sheaf, namely OZ(,) (regarded as a sheaf on X ) in highest degree 0. The complex IC+(E, s) in this situation has the only non-trivial cohomology sheaf in the highest degree r, and this sheaf is the restriction det ElZ(,)= ArEIZ(,) regarded as a sheaf on X.

Sketch of the proof. We shall prove only (i), the proof of (ii) follows the same ideas but is more technically involved; see e.g. [Ful, GHl]. If V is a vector space and v E V is a non-zero vector then the differential in the exterior algebra A*V given by Av is exact. Indeed, we can include v in some basis as a first vector, and then both the kernel and the image of the differential are spanned by monomials containing v. Therefore the dual differential i, in A*VV is also exact. Applying this to our situation we see that if 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 E Z(s) then, trivializing E near x, we represent s as a collection of functions (fl,. . . , fr) E Or. Then the last differential in both complexes has a form Or + 0, ( ~ 1 ,... ,ur) H C ui fi. If s vanishes a t x then all f i = 0 0 and this map is not surjective. Let now X c P(V) be a smooth projective variety. Then we can apply Theorem 5.17 to the jet bundle E = J(L), where L = Ox(1). Combining Proposition 5.16 and Theorem 5.17 we get the following theorem. Theorem 5.18 ([GKZZ]) For any f E VV let j = j ( f ) denote its first jet. Then f represents a point in X* if and only if any of the following complexes of sheaves on X is not exact:

5.3 Cayley Method

99

5.3.2 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 the Cayley results. In topology, determinants of complexes were introduced in 1935 by Reidermeister and Franz [Fra]. They used the word 'torsion' for the determinant-type invariants constructed. In this section we give only the definitions that are necessary for the formulation of results related to dual varieties. More details could be found in [GKZB, 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 AdimVv 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 t W -+ 0 we have a natural isomorphism Det V = Det U 8 Det W. Suppose now that V = h @ Vl is a finite-dimensional supervector space. Then, by definition, Det V is set to be Det Vo 8 (Det Vl)'. Once again, for any exact triple of supervector spaces 0 4 U t V -+ W -+ 0 we have Det V = Det U 8 Det W. For any supervector space V we denote by p the = fi, pl = h . Clearly, we have a natural new supervector space given by isomorphism Det p = (Det V)'. (5.2) Now let (V, d) be a finite-dimensional supervector space with a differential d such that d h c fi, dfi c h, d2 = 0. Then Kerd, Imd, and the cohomology space H(V) = Ker d/ Im d are again supervector spaces. We claim that there exists a natural isomorphism

6

Det V cz Det H(V).

(5.3)

Indeed, from the exact sequence

we see that Det V E Det (Ker 8) 8 Det(1m 8)'. From the exact sequence

-

we deduce that Det H(V) Det(Ker d) 8 Det(1m 8)'. This gives (5.3). In particular, if d is exact (i.e. H(V) = 0) then we have a natural isomorphism Det V N k .

(5.4)

100


Let us fix some bases {el,. . . ,edimv,) in Tro and {el,. . . , e&imVl> in Vl. Let {f:, . . . , f&, Vl) be a dual basis in KV.Then we have a basis vector (el A

.. . A edimv,) 8 (fi A .. . A fiimvl) E Det

Therefore, if (V, a, e) is a based supervector space with an exact differential, then by (5.4) we get a number det(V, a, e) called the Cayley determinant of a based supervector space with an exact differential. If we fix other bases {El, . . . ,Edim vo) in & and {El, . . . , ELirn vl) in K then, clearly, e t det(V, d, e), det(V, a, E) = det ~ ~ ( dAl)-l

(5.5)

where (Ao,Al) E GL(K) x GL(K) are transition matrices from bases e to bases E. One upshot of this is the fact that if bases e and E are equivalent over some subfield ko C lc then the Cayley determinants with respect to these bases are equal up to a non-zero multiple from 50. The homogeneity condition (5.5) can be reformulated as follows. For any based supervector space (V,a, e) with an exact differential and any (Ao,Al) E GL(V0) x GL(K), we can consider a new based supervector space (V, A. a, e) with the exact differential A. 8 given by A.dlvo = A l . d . ~ , l , A.dlvl = A 0 . d . ~ y 1 . Then we have det(V, A . a, e) = (det AO)-' det A1 det(V, a, e). In particular, it is easy to see that the following formula is valid:

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

+ dim Im do = dim Im dl + dim Ker dl = dim K = n.

Therefore both do and dl are represented by (n x n) matrices Do and Dl. For any subsets I,J c B = {I,. . . ,n) we denote by D i J and D f J the submatrices of Do and D l formed by columns indexed by I and rows indexed by J. Then Il c B such that the submatrices it is easy to see that there exist subsets lo, D P \ ' ~ ) ' ' ~ ,D $ ~ \ * ~ are ) " ~invertible. In particular,

The formula (5.5) implies that det (V, a, e) = det D ~ ~ " o )(det "~

)'I0 )-I.

(5.7)

This formula gives an explicit matrix description of the Cayley determinant.

5.3 Cayley Method

101

Now we consider a finite complex of finite-dimensional vector spaces

Then we can define the finite-dimensional supervector space V

Vo=

@ i ~ mod 0 2

vi,

&=

$ i=l mod 2

=

Vo @ Vl,

vi,

with an induced differential d. In particular, all previous considerations are valid. Therefore, if the complex (V0,d) is exact and there are some fixed bases {el, . . . ,eii, v i) in each component Vi, then we have the corresponding Cayley determinant det(Va,d,e) G k*. For example, if L and M are based vector spaces and A : L -+ M is an invertible operator then the complex A 0 -+ L -+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

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 (5.2) we have det(V0[m],d , e) = (det(V0[m],8,e))(-llm In the formula (5.8) we may suppose that all Bi are finite sets. Therefore Di is a matrix with columns indexed by Bi and rows indexed by Bi+l. For any subsets X c Bi and Y c Bi+l we denote by (Di)xu the submatrix in Di with columns from X and rows from Y. A collection of subsets Ii C Bi is called admissible if lo= 0, I, = B,, and IBi \ Ii 1 = IIi+l1 for any i = 0, . . . ,r - 1, and finally the submatrix ,I++,is invertible. The following theorem follows easily from (5.7). (Di) (Bi\17)

Theorem 5.19 ( [ G K Z B ] ) (i) Admissible collections exist. For any admissible collection we have

(ii) Let {Ii)be an admissible collection. Denote by matrix (Di)(Bi\Ii),li+l. Then

Ai the determinant of the

102


In particular, this theorem (or formula (5.6)) implies the following corollary Corollary 5.20 For any based exact complex (V', d, e) we have det(V0,Ad, e) = Ax> 0. For any coherent sheaf 3 on X , we write

3(1) = 3 8 O x @ ) ,h i ( 3 ) = dim H'(x, 3),x ( F ) = x(-l)W(3). The number ~ ( 3 is ) called the Euler characteristic of 3. It follows from the Riemann-Roch-Hirzebruch theorem [Hir] that x(F(1))is a polynomial in 1. This polynomial is called the Hilbert polynomial of 3 and is denoted by h F ( l ) .For 1 >> 0 , the higher cohomology of 3(1) vanish and we have

Simple calculations show that Theorem 6.10 can be reformulated as follows. Recall that T x is the tangent bundle of X and 0; = AiT2 is the bundle of differential i-forms on X . Theorem 6.11 ([GKZ2]) For any 1 E Z we have

116

6 Degree of the Dual Variety

This theorem was generalized in [GK]to handle the case of positive def X . For any 1 E Z consider the polynomial fl (q) in a formal variable q dim X+l

fi(4) =

( h n i f i (2 i=O

+ 1 ) + h ~ i - (i1 +~1 )~) qi,

where all polynomials that do not make sense are equal to 0. Clearly fi(q) is also a polynomial in I .

Theorem 6.12 ([GK, GKZ21) Let 1 E Z+ be any non-negative integer. The codimension of X * equals the order of the zero of fl (q) at q = 1. Let this order be p and let f l ( d = a , ( W - d P+ O((1 - dP+l).

Then

6.2.2 Lascoux Formula If the coherent sheaf .F is supported on the subvariety of codimension n, i.e. on the set of points {xi), then Fziis a vector space of dimension ni and the corresponding element of A n ( X ) is just C nixi. The corresponding element of H o ( X ,Z ) = Z is just C nil and therefore the homomorphism

is obtained by taking the Euler characteristic. It follows that if a E K ( X ) is an element of generic rank p then

S,

~n( a )=

x (An( a - O, - n + 1)[ O x ] .)

Theorem 6.13 ([Las]) For any p E Z ,

Proof. Indeed, by Theorem 6.1, we have

By the calculations above, we have

From the exact sequence

6.2 Formulas Related to the Cayley Method

117

and therefore deg A x = x (An([Ll+ [Ll[ T ' I - 2[0x1) . It remains to tensor the right hand side with [CVIn. [LIP. This is possible, because if F is supported on a finite set. Then, obviously, x([F]. M ) = x([F]) 0 for any line bundle M.

Varieties with Positive Defect

In this chapter we study varieties with positive defect. In Sect. 7.1 we focus on beautiful theorems of Ein about the normal bundle of a generic contact locus. Since this locus is a projective subspace, it is possible to use the machinery of vector bundles on projective spaces. We prove Ein's theorem that this normal bundle is symmetric and uniform, which explains among other things a parity theorem of Landman. We introduce the Beilinson spectral sequence and use it to calculate the normal bundle to a generic contact locus in small dimensions. Finally, we study dual varieties of scrolls and prove a theorem of Ein that a variety of defect at least 2 is a scroll if and only if the normal bundle to a generic contact locus splits. In Sect. 7.2 we follow [IL] and discuss linear systems of quadrics of constant rank and how they are related to dual varieties via the second fundamental form. In Sect. 7.3 we prove a theorem of Beltrametti, Fania, and Sommese that relates the defect of a projective variety and its Mori-theoretic characteristic called the nef value. We give a brief survey of necessary results from Mori theory. We finish by giving a classification of smooth varieties of positive defect up to dimension 10 obtained by many authors and initiated by Ein. Finally, in Sect. 7.4 we use this connection with Mori theory to classify all flag varieties with positive defect. This approach was developed by Snow in contrast with the original proof of Knop and Menzel that used the Katz dimension formula.

7.1 Normal Bundle of the Contact Locus 7.1.1 Ein Theorems Suppose that X c IPN is a non-degenerate smooth projective variety, dim X = n. For any hyperplane H C IPN the contact locus (X n H),i,g is a subvariety of X consisting of all points x E X such that the embedded tangent space 5?z,X is contained in H. One can use the Jacobian ideal of X n H to define a scheme structure on the contact locus. This scheme could however be not

120

7 Varieties with Positive Defect

reduced. Clearly the contact locus is non-empty if and only if H belongs to the dual variety X * . By Theorem 1.18, if def X = d then for any H E (X*),, the contact locus (H n X),in, is a projective subspace of dimension d, and a union of these projective subspaces is dense in X . In this section we study further properties of the contact locus.

Theorem 7.1 ( [ E l ] )Suppose that q is a generic point of X , that H is generic tangent hgperplane of X at q, and that L = lPd is the contact locus of H with X and d > 0. Then there exists a symmetric isomorphism NL,X N X J ( 1 ) .

Proof. Let sh be the section of O x ( 1 ) defining H n X and let IL c O X be the ideal sheaf of L in X . We choose a local coordinate system Isl,.. . , x,) of X near p. We may assume that I L is generated by X I , . . . , xn-d. Since L c H n X , we can write the power series of sh in the following form:

where f i , . . . ,fn-d are power series in variables xn-d+l,. . . ,x,. But ( H n X),ing = L, and therefore (dsh/dxi)lL= 0 for i = 1,.. . ,n - d. Hence

Thus sh E 12(1). Therefore sh gives a section of

and induces a map from NL,X to N L , x ( l ) .It remains to show that this map is an isomorphism. We can write g i j = aij h i j , where the aij are constants and the hij are power series without the constant term. Now

+

n-d

It follows from Theorem 3.1, using Sard's Lemma for algebraic varieties, that the tangent cone of the hyperplane section H n X at p is a quadric hypersurface n-d

C xizjaij is a quadratic form of rank n - d i,j=l and this is also the equation for the quadric hypersurface in P ( N L J l p ) induced 0 by She Recall that by a theorem of Grothendieck any vector bundle on JP1 has a form e i O ( a i ) for some integers ai (see e.g. [ H a l ] ) .

of rank n - d in T ~ ,Therefore ~ .

7.1 Normal Bundle of the Contact Locus

121

Definition 7.2 A vector bundle E on a projective space iFN is called uniform if the restriction ElT is a fixed vector bundle $iO(ai) for any line T c iFN. Theorem 7.3 ([El]) Suppose that q is a generic point of X , that H is generic tangent hyperplane of X at q, and that L = IPd is the contact locw of H with X and d > 0. Then N L , is ~ a uniform vector bundle and

for any line T i n L. Proof. Consider the exact sequence

where NL,PNIT = (N - d ) O ~ ( l )Suppose . that

Then exactness implies that all ai 5 1. Using the isomorphism between N L , ~ and Nx,X(l) we observe that each ai 2 0. Therefore

and NLX is a uniform vector bundle. This implies that

0 and the theorem is proved. As a formal consequence of Theorem 7.3 we get the following parity theorem that was first proved by A. Landman, using the Picard-Lefschetz theory (unpublished) :

Theorem 7.4 ([El]) If def X > 0 then dimX

-- def X

mod 2.

Example 7.5 ([GH]) Suppose that X c IPN is a smooth non-linear surface. Then X* is a hypersurface. Indeed, if def X > 0 then def X is even by Theorem 7.4. But X obviously can not contain an even-dimensional projective subspace IPdef X . Example 7.6 Suppose that X C PN is a smooth projective nonlinear variety with dimX 2 2. Then def X 5 dimX - 2. Indeed, it is clear that def X dimX - 1. Therefore, by Theorem 7.4 we have def X 5 dimX - 2. Moreover, it can be shown [El] that def X = dimX - 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 PN linearly (cf. Theorem 5.14 and remarks after it).

Proof. Consider projections pl ,pz : PN x PN -+ IPN and abbreviate A El B = pTA 8 paB for any bundles A, B over PN. Consider the bundle

over PN x PN. Using the obvious exact sequence

and Kiinneth formula, we get

~


125

Therefore I has the canonical section s that corresponds to the identity in Hom(CN+l , CN+l). More explicitly, let v, w E CNfl \ (0) and let x, y E PN be corresponding lines. Then

is defined as s(x, y)(Av) = Av mod Cw. Clearly, s(x, y) vanishes if and only if x = y, i.e. the zero scheme of s is the diagonal A c PN x PN. Consider the corresponding exact Koszul complex

which can be rewritten as

By tensoring this exact sequence with pyE, we get the complex C* of sheaves

with cohomology

{

H y c * ) = p ; E l ~ for q = 0 0 otherwise. Consider the complexes of Abelian groups L*>jcalculating the cohomology of every individual C j (for example, the Cech complexes with respect to an appropriate open covering of PN x PN). The differential in C* makes this collection of complexes into a double complex La*.We define the i-th hyperdirect image Ri(p2),(C*) as the i-th cohomology sheaf of the total chain complex associated to (p2)*(La*). There are two spectral sequences 'E and "E associated to the double complex (p2), (La*)

Since pq E2 = RP(p2)*(Hq(C*))=

11

we have

{ E0

~"p2)*(~= *)

E forp=q=O 0 otherwise, for i = 0 otherwise.

However,

It follows that 'E is the spectral sequence we were looking for.

126


Definition 7.16 A monad over Pk is a complex

of vector bundles which is exact at A and C, such that Im(a) is a subbundle of B. The vector bundle E = Ker(b)/Im(a) is called the cohomology of the monad.

7.1.4 Planes in the Contact Locus Now we return to our situation: X C PN is a non-degenerate smooth projective variety of positive defect d = def X and dim X = n. Let L = ( H nX),i,, be a generic contact locus. We would like to use the Beilinson spectral sequence for the study of the bundle NLlx. Let us describe the restriction of N;,X to a plane in L.

Theorem 7.17 ([Mun3]) Let X c PN be a variety of dimension n with positive defect d > 2. Let H c X* be a generic tangent hyperplane and L the contact locus of X with H . Let R c L be a plane. (i) Nz,XIR has the following structure:

where M is the cohomology of a monad

(ii) h1(NzIX(- 1)IR) is a n even number

Proof. (i) The symmetric isomorphism between N;,X and NLTx(l)produces a symmetric isomorphism between the corresponding restrictions to R:

Since NL,XIRis globally generated then, as in Theorem 7.8, we have the splitting Nz,XI= ~ M @ [OR(-l) @ o ~ ] @ ~ ~ ( ~ : ~ Using the Serre duality, we see that the first term

Erq= H q ( M( p ) ) @ K p ( - p ) of the Beilinson spectral sequence of the vector bundle M is:

~~~)


127

This implies that Egg = E z and hence, by the convergence of the Beilinson spectral sequence to M, the latter is the cohomology of the monad corresponding to the second row in Ey4. (ii) Since M E M "(-1) and

by [BH] the isomorphism between M and M*(-1) extends to a morphism between the monads

By [BH],H is a symmetric isomorphism. It remains to notice that a symmetric isomorphism between T:(l)%nd TR(-2)"xists if and only if k is even. Indeed, this follows from two facts. Firstly, an isomorphism between Tg(1) and TR(-2) is unique and skew-symmetric (it is given by a usual product of 1-forms). Secondly, if a block matrix

is symmetric, Ic is odd, and each Mij is a skew-symmetric (2 x 2)-matrix, then it is easy to check that det M = 0. 0

Theorem 7.18 ([Mun3]) Let X c IPN be an n-dimensional variety with positive defect d. Let H c X * be a generic tangent hyperplane and L the contact locus of X with H. Then h0(NL,x) 2 n. If ~ ' ( N L J ) = n then Tan(L, X) is a linear subspace. Proof. Take the exact sequence

Let s = ~ O ( N : , ~ ~ ( ~ ) JBy L ) .Theorem 5.2, we have dim(Tan(L, X)) = N - s and this implies our theorem, because dimTan(L, X ) = n+d by Theorem 4.24 (otherwise X c Sec(L, X ) = Tan(L, X ) c H

128


0 which contradicts the hypothesis of X being not degenerate). These results can be used to get a list of all possible candidates for the restriction E of N l , X to a generic plane R in L. This classification was obtained in [MunS]. a

If rk E = 2 then

E = OR $ OR(-1). a

If r k E = 4 then

0g2$ OR(-I)@~ = {T;(l)@?

where E3 appears in an exact sequence of the form

a

If rk E = 8 then

where E4 appears in an exact sequence of the form

a

If rk E = 10 then

where E5,E6, and E7 appear in the following exact sequences


129

Example 7.19 Suppose that X = Gr(2,C2r+1) in the Pliicker embedding. Then dimX = 2(2r - I), def X = 2, and Nl,X = T&(1)@2r-2. Example 7.20 Consider the spinor variety X = S5 in its minimal linearly normal embedding. Then d i m 1 = 10, def X = 4, and N l , X = 0&(2). Let XI be a generic hyperplane section of 95. Then dimX1 = 9, def XI = 3, and Nl,x, = 0&(2) @ 0&(1). Let XI1 be a generic section of S5 by two generic hyperplanes. Then dim X" = 8, def X" = 2, and N,Yxr, = 0p2 @ T&(1)B2 @ 0JP2(-l). 7.1.5 Scrolls

The following theorem is well-known:

Theorem 7.21 Let X C PN be an n-dimensional scroll, i.e. a projective bundle Py (E) over a smooth variety Y such that all fibers are embedded linearly, and let dimY = m. Suppose that n 2 2m. Then def X = n - 2m and Nx,X splits as a sum of line bundles. Proof. Take a general x E X and a general H E X* such that FX,xc H . We denote by p the map p : X --+ Y. The fiber p-lp(x) = Pn-m must be contained in Fx,x. We can take m analytic arcs yi (t) such that yi(0) = p(z) and such that their tangent vectors span the linear space of directions of T y , y These produce m l-parametric families Fl of linear spaces of dimension n - m. For any point y E pF1p(x), any choice of analytic arcs cri(t) E F; with ai(0) = y verifies that ai(0) is a set of linearly independent vectors. The hyperplane H cuts the general member of each one of these families in a linear space of codimension lower than or equal to 1: F; fl H = G:. Then the of codimension limit position on p-lp(x) of these G: are m linear spaces lower than or equal to one in p-lp(x). Since n f must be contained in the contact locus of X with H , we have shown that defX 2 n - 2m. If n = 2m, see Theorem 5.14. If n > 2m, then the general hyperplane section of X is a scroll XI = Py(Et), where dim El = dim E - 1. We proceed by induction using Theorem 5.3. We see that def X = n - 2m and L c p-lp(x). Now N L , splits ~ by Example 7.9. 0

fi

Theorem 7.22 ([E2, Mun31) Let X C PN be an n-dimensional variety with defect def X > 1. If NL,X splits as a sum of line bundles then X is a scroll.

y.

Proof. Let m = Then NL,X= OFm @ O L ( ~ ) @ Take ~ . a general x E L, and let Zx be the ideal sheaf of x in L. Since

the Hilbert scheme of k-planes in X through x has a single irreducible component 3.1 through the point corresponding to L, and dim3.1 = km. Take the universal family


130

I = { ( y , ~ )E X ~ 3 - 1 s)%x. 1 ~ ~ Then dimp(I) = km

+ k - dimp-l(y)

with y a general point of S and S a general point of 'FI. Since a general deformation of a split bundle on IPk is also a split bundle, we observe that Ns,x = OFm @ OS(l)'m. On the other hand, take the ideal sheaf of the line (x, y) in S. Then

+

Then p(I) has dimension k m and contains a family of dimension km of k-planes through x. This family has a maximal possible dimension, which means (see e.g. [Rog]) that D = p(I) is a linear space. Hence N D t x = OFm for any k-plane S in our family, i.e. ND,x is a uniform vector bundle of type = OFm, e.g. by [E3]. Hence the claim follows (0,0,. . . ,0). Therefore, 0 from Theorem 7.23 below.

Is

Theorem 7.23 ([E2]) Let X be a projective n-fold in I P N . Assume that there is an m-plane Do in X such that ND,,x = O$r-m and n 5 2m - 1. Then X = Py(F)is a scroll on an (n - m)-fold Y. Theorem 7.22 and Corollary 7.30 imply the following theorem.

Theorem 7.24 ([E2]) If X is a smooth n-dimensional projective variety in IPN and def X = k > n/2, then X is a projective bundle X E I 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.

7.2 Linear Systems of Quadrics of Constant Rank Definition 7.25 Let V = Cm and W = Cn. We take a linear subspace

A

c Hom(V, W)

(resp. A

c s 2 v V , resp. A c A2vV).

A is said to be of constant rank r if, for all non-zero f E A, rank f = r. We define numbers 1(r,m , n) (resp. c(r, m), resp. X(r, m)) as the maximal possible dimension of a linear subspace in Hom(V, W) (resp. S2VV,resp. A2VV) of constant rank r. Given a linear subspace A c Hom(V, W) we can also consider A as a linear subspace ACof S2(VV@ W) or a linear subspace AX of A2( v V@ W). In matrix form, for any matrix a E A we associate a symmetric matrix a C E AC or skew-symmetric matrix a X E AX of the form

7.2 Linear Systems of Quadrics of Constant Rank

131

If A has constant rank r then AC and AX have constant rank 2r. AC and AX are called doublings of A.

Proposition 7.26 ([IL]) (i) If0 < r I m 5 n then l(r, m, n) 2 n - r (ii) If r 2 2 is even thenc(r,m) 2 m - r + 1 and X(r,m) 2 m - r + 1.

+ 1.

Proof. Let XT-l c P(Hom(CT,Cn) be the space of matrices of rank less then r . Then c ~ d i m X , - ~= n - r 1. Thus we can find an (n - r 1)dimensional subspace A c Hom(CT,Cn) of constant rank r and (i) follows. The doublings of A produce ( n - r 1)-dimensional subspaces AC C S2CT+n 0 and AX c A2CTfn of constant rank 2r. This implies (ii). Any linear map $ : A -+ Hom(V, W) can be viewed as an element of

+ +

+

A" 8 Hom(V, W) = HO(P(A), OqA)(1)) @ Hom(V, W). Thus to give a linear map A 4 Hom(V, W) is the same as to give a vector bundle map $ : V @ @(A) W '8 (%(A) (1) on P(A). Let K , N , and E be the kernel, the cokernel, and the image of $. Then it is easy to see that is an embedding of constant rank r iff K , N, and E are vector bundles of ranks dim V - r, dim W - r , and r iff any of K, N , and E is a vector bundle of rank as above. It follows that embeddings A c Hom(V, W) of constant rank are in one-one correspondence with pairs of exact sequences of vector bundles

+

Similarly, all embeddings A C S 2 V (resp. A c A2V) of constant rank are in one-one correspondence with exact sequences of vector bundles

where E is a rank r vector bundle such that E E E*(1) and this isomorphism is symmetric (resp. skew-symmetric).

Theorem 7.27 ([IL]) Let A C S 2 V or A c A2V be a subspace of constant rank r with dimA 2 2. Then E is a uniform vector bundle of splitting type t~ O;j2(1). In particular, r is even.

0i[2

Proof. The proof is by the same argument as in the proof of Theorem 7.3. The parity of r was known classically as a consequence of the KroneckerWeierstrass theory giving a normal form for pencils of symmetric matrices of 0 bounded rank; see [Gan, HP, Me].

132


Theorem 7.28 ([IL]) A smooth n-dimensional projective variety X c PN with d = def X > 0 determines a (d + 1)-dimensional linear subspace A c s2cN-1-d ith the associated short exact sequence

where H E X * is a generic tangent hyperplane and L = (X n H),i,, contact locus.

is the

Proof. Recall that L is a projective subspace of dimension d. We have the following exact sequence of vector bundles

which holds because H is tangent to X along L. Dualising and twisting by O L ( l ) we get 0 -+ F V ( l ) -+ N,V,,(l) -+ N,VIX(1)-+ 0. Recall that N L , x rr N,Vlx(l) by Theorem 7.1. It remains to notice that NL,H N ( 3 ~ ( 1@) c ~ - ~ - ~ .

Theorem 7.29 ([El, IL]) Suppose that A subspace of constant rank r . Let

C

S 2 V is an (1

+ 1)-dimensional

be the corresponding exact sequence of vector bundles. If r 5 1 then E is isomorphic to @ 0;;:)( 1 ) unless r = 1 = 2 and E = Tpl(- 1).

4;;)

Proof. Indeed, E is uniform by Theorem 7.27. Therefore, if r 5 1, then E either splits into a direct sum of line bundles or E is isomorphic to Tpl(a) or T$ ( a ) by [Sa]. If E splits then E rr 0;;;)@ ( 1 ) by Theorem 7.27. Suppose now that E is isomorphic to TPl(a) or T$(a). Since E rr E V ( l ) , the first Chern class c l ( E ) is equal to (1/2)H, where H is the class of the hyperplane section. Computing the first Chern classes of T p ( a ) and Tg ( a ) we conclude that E is isomorphic either to Tpz(-1) or to T$ (2). These bundles 0 are in fact isomorphic.

4;:)

Corollary 7.30 ([El, IL]) Suppose that X c PN is a smooth projective variety. Let L be a generic contact locus. If def X 2 n / 2 , then (dimX-def X ) / 2

NL,X = O L

(dimX-def X ) / 2

@ QL

(1).

Proof. Let dimX = n and def X = k . By Theorems 7.29 and 7.28, we see that N L , x is isomorphic to either (3p-k)'2@ 0 p - " ' ~ ( 1 ) or TPz(-1). However, in the last case n = 4, Ic = 2, and therefore X is a projective bundle P c ( F ) over a curve C by Example 7.6. Then L is embedded as a Zplane in a fibre f of PC ( F ). Consider the exact sequence


133

Since NL,f = OL(l) and Nf,x = Of, this exact sequence reduces to

0 It follows that NL,X= OL @ OL(l). The following theorem was proved in [W]. Weaker results (but similar arguments) first appeared in [Sy].

Theorem 7.31

([w)Suppose that 2 5 r 5 m 5 n. Then

(i) I(r,m,n) I m + n - 2 r + 1 . (ii) l(r, m, n) = n - r 1 if n - r (iii) l ( r , r + 1 , 2 r - 1) = r + 1.

+

+ 1 does not divide (m - l)!/(r

-

I)!.

Sketch of the proof. Let A c Hom(V, W) be an 1-dimensional linear subspace of constant rank r , where dimV = m, dim W = n, and m n. Let

>

0 +K

+ O P ( ~@ )

V + E + 0 and 0 -+ E -+ O P ( ~ ) ( @ l )W

-+

N

-+

0

be the corresponding exact sequences. Then c(K)c(E) = 1 and c(E)c(N) = (l+H)n, where H is the class of the hyperplane section. Thus, c ( K ) ( ~ + H= )~ c(N). If n - r 1 5 i 5 I - 1 then ci(N) = 0 and, looking at the coefficient of H i , we get

+

m-r

j=O

(7)

= 0 if i < 0 or i where cj(K) = kjHj and this system of linear equations is

If I = m

> n. The coefficient matrix of

+ n - 2r + 2 then this is a square invertible matrix with determinant

Thus ko = 0 which is a contradiction to ko = 1. This proves (i). Proofs of (ii) and (iii) can be found in [W]. By Proposition 7.26 (i) we n - r 1, and so only the opposite inequality should be have l(r,m,n) 0 proved. Another possible way of proving Theorem 7.31 (i) is to use the following theorem of R. Lazarsfeld (its application is immediate):

>

+

134


Theorem 7.32 ([Lal]) Let X be a projective variety with dimX = 1. Let E and F be vector bundles of ranks m and n, respectively. Suppose that EV@ F is ample and there is a constant rank r vector bundle map E -+ F . Then 1im+n-2r. The (partial) symmetric analogue of this theorem was found in [IL]. Theorems 7.32 and 7.33 (and their proofs) are analogous to results related to the problem of the non-emptiness and the connectedness of degeneracy loci, see [Lal, FL1, HT, HT1, Tu, IL]. Theorem 7.33 ([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 o n X . Suppose that S2EV @ L is an ample vector bundle and that there is a constant even rank r symmetric bundle map E --t EV@ L. Then m 5 n - r . Proof. Consider the projective bundle n : P(E) -+ X. The symmetric bundle map E -+ EV8 L gives a section

which pulls back to

( 2 )0qE)(2). Since ~ * 0 p ( ~ ) (=2S There is a natural map ~ * n * O p ( ~ ) -+ ) 2~' this gives after tensoring with n*L a map

Let

t = u 0 T*S E H'(P(E), Op(E)(2)@ n*L). Let Y c P(E) be the zero locus of t. Let x E X. The section t at [v]E P(E(x)) is the linear map

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 ny : Y -t X over each x E X is a rank r quadric hypersurface in P(E(x)). We claim that @yE) (2)@n*( L ) is ample on P(E). Indeed, since S2E V @ Lis is ample on P(S2EV@L).Clearly, P(S2EV@L) e ample on X , C)nn(SzEv,g,L)(l) P(S2EV)and Op(S~E~g,L)(l) 21 O p ( S ~ E ~ )@ ( lu*L, ) where u : P(S2EV)--t X ) is ample on P(S2EV). is the natural projection. Therefore O p ( S ~ E ~ )@( lu*L The second Veronese map gives an inclusion i : P(E) c P(S2EV)such that n = i o a. Then i*Op(saEv)(l) = Op(E)(2). Hence i*(0P(S2EV)(l)8 u*L) N O P ( ~ ) ( 2@) n*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:


135

Hi(P(E)\Y,@) = O f o r i > m + n . Let K denote the kernel and F denote the image of the map E -+ EV@ 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 E FV@ L. We have a natural map

given on the fibers by the linear projection centered at P(K(x)) c P(E(x)). Then p is a @n-r-fiber bundle. L determines a hypersurface Z C P ( F ) such The isomorphism F FV18 that, if p : P(F) -+ X is the natural projection, then the fiber of plz : Z -+ X over each x E X is a smooth quadric in P(F(x)). Now P(K) c Y and so p restricts to a @n-T-fiber bundle P(E) \ Y -+ P(F) \ Z. Thus

--

c)

and, by Lefschetz duality, this is isomorphic to H ~ ( ~ + ~ ( q -F ~ ) , 2, ) - ~Hence H'(P(F),z,@) = O f o r i < 2 r + m - n - 2 and then using the long exact sequence of the pair (P(F), Z) we conclude that H~(z,@) E H~(P(F), @) for i 5 2r

+ m - n - 3.

Both bundles p : P ( F ) -+ X and plz : Z -+ X have smooth fibers and are defined over a smooth simply-connected variety. Therefore, by Deligne' theorem [GHl], the Leray spectral sequence for p and piZ degenerates at the E2 term. It follows that

and He(Z, @) = He(X, @) @ He(Q, @), where Q c Pr-I is a smooth quadric. Let bj (B) = dim H j ( B , @) be the Betti numbers. Then it is well-known that bi(PT-l) =

{ 01

foreveni1O5i0

c IF'(s2vV) is of constant rank r ) = m - r

Example 7.35 Theorems 7.28 and 7.34 combined furnish a new proof of the formula dim X * > dim X (Theorem 4.25 (ii)).

7.3 Defect and Nef Value 7.3.1 Some Results from Mori Theory Divisors Let X be a smooth projective variety. If H is a very ample Cartier divisor on X (a hyperplane section in some projective embedding) then a pair (X, H) will be called a polarized variety. A divisor D on X is called semi-ample if the linear system lmDl is basepoint-free for some m >> 0. Let Kx be a canonical divisor. We denote by Div(X) a free abelian group of Cartier divisors on X . We D2 if Dl and D2 are linearly equivalent (differ by a principal write Dl divisor). Pic(X) is the group of algebraic line bundles on X . It is naturally isomorphic to a quotient group of Div(X) modulo principal divisors. For any D E Div(X) the corresponding line bundle is denoted by O(D). N

A 1-cycle on X is a finite sum of irreducible curves Cic X with integer multiplicities ni. Z is called efective if all ni 0.

>

The intersection product If D is a Cartier divisor on X and C c X is an irreducible curve then an intersection product D C E Z is defined as deg g*O(D), where g : -+ C is a normalisation. This product extends by linearity to a %valued pairing of free abelian groups Div(X) and a group of 1-cycles. 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 (written Zl E 22) if Z1 - Z2 is numerically trivial. Dually, two Cartier divisors are D2) if D l- D2 is numerically trivial. numerically equivalent (written Dl The group N . ( X ) of Cartier divisors modulo numerical equivalence is a free abelian group of finite rank p(X) called a Picard number of X . The group NF(X) of 1-cycles modulo numerical equivalence is also a free abelian group of rank p(X).

-

7.3 Defect and Nef Value

137

T h e cone of effective 1-cycles We consider two real vector spaces of dimension p(X):

These vector spaces are naturally dual to each other by means of the intersection product. The following is the main combinatorial object of the Mori theory: Definition 7.36 0

0

The cone of eflective 1-cycles NEl(X) c Nl(X) is a cone generated by classes of effective 1-cycles. (x) is its closure in Nl (X). The positive part NE+ (X) of (x) is defined as

m1

Nef divisors

>

For D E N1(X), we say that D is nef (or numerically eflective) if D . Z 0 for any Z E NE1 (X) (or equivalently for any Z E NE1 (X)). This gives a closed cone NE1(X) in N1(X) dual to NEl(X). For example, if D is an ample divisor then C D > 0 for any curve C C X and therefore D is nef. The following is a useful Kleiman's criterion: Theorem NEl(X)'

7.37 ([K16]) D is ample if and only if D . Z > 0 for any Z E # 0.

In other words, ample divisors correspond to the interior of NE1(X). All introduced cones are non-degenerate (are not contained in any hyperplane).

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 N1(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

m1(X) such that

Kx . R < 0 (Kx- Z < 0 for any non-zero Z 0

m1

R is a face of (X), i.e. if Zl Z1 and Z2 are in R.

E R);

+ Z2 E R for any Z1, Zz E

(X) then

138


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:

1, the ~ 0), then D is semi-ample. divisor a D - K x is nef and big (2.e. ( a D - K x ) > One of the most useful forms of the Contraction Theorem is as follows.

m1

Theorem 7.40 Let R be an extremal ray i n (X). Then there exists a normal projective variety Y and a surjective morphism contrR : X --+ Y with connected fibers such that c o n t r ~contracts a curve C t o a point if and only if [C]E R. Proof. Indeed, let D E NE'(X) be such that D . R = 0 and D . Z > 0 for any Z E m l ( X ) \ R. It follows from the Mori Cone Theorem that we may assume that D is a divisor. (In fact, usually this theorem is proved as 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 implies that for a >> 0 the divisor a D - K x is ample. 0 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 Kx is not nef.

Definition 7.41 T

= min{t E R

I Kx

+ t H is nef}

is called a nef value of (X, H). If X is fixed then T 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 < m. Notice also that T is a nef value of (X,H ) if and only if K x TH is nef, but not ample. The following theorem is known as the Kawamata Rationality Theorem:

+

Theorem 7.42 ([Kaw]) T is a rational number. The next theorem is a particular case of the Contraction Theorem


139

Theorem 7.43 Suppose that r is a nef value of (X, H ) . The linear system Im(Kx 7H)I is base-point-free for m >> 0. It defines a regular morphism sP : X -t Y onto a normal variety Y with connected fibers called a nef value morphism.

+

Length of extremal rays If R is an extremal ray, then its length l ( R ) is

l ( R ) = min{-Kx

. C / C is a rational curve with

[C]E R ) .

The motivation for this definition is to give an estimate of the dimension of the locus of extremal rational curves in an extremal ray. The idea of the proof of the following theorem is, basically, due to Mori [Morl]. We will use the same argument several times in this section but the details will be left to the reader.

Theorem 7.44 ([Wis]) Let C be a rational curve with its class i n an extremal ray R, such that -Kx. C = 1 ( R ) .If Q is a smooth point o n C then the locus of points of curves that belong to R and pass through Q has dimension at least 1(R)- 1. Proof. By the local deformation theory the space of deformations of the morphism f : P1 -+ X , f (P1)= C , has dimension at least

Since we want to fix the point Q , we have dimX more restrictions. Therefore the space of deformations of the morphism f : P1 t X fixing the point Q has dimension at least -Kx . C = l ( R ) . The group of automorphisms of P1 fixing a point x P1 has dimension 2, and 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, and 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 D. Passing to infinite points D\ D, we get limit positions of these curves that are irreducible and reduced, since otherwise we obtain a contradiction with minimality of -Kx . C for rational curves from the ray R. Therefore there exists a ruled surface S and a morphism F : S -t X such that Cl = F - l ( P ) and Cz = F - l ( 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

140


C;

< 0, C: < 0, and CI . C2 = 0,

(7.2)

the last equality holding because Cl and Cz do not intersect. But this is impossible: a Picard number of a ruled surface is 2 and therefore (7.2) contradicts 0 the Hodge index theorem [Hal].

Fano varieties

A smooth variety X is called a Fano variety if -Kx is ample. Its index is defined as follows: r(X) = max{m

> 0 ImH

N

-KX for some H

E Pic(X)).

+

It is well-known that r(X) 5 dim X 1. Indeed, suppose that Kx = mH, where - H is ample and let dimX = n. The Poincare polynomial x(vH) has at most n zeros, and therefore x(vH) # 0 for some 1 v 5 n 1. But x(vH) = fhn (X, vH) by the Kodaira vanishing theorem and hn (X, vH) = h O ( X Kx , - vH) by the Serre duality. Therefore m n 1. The following theorem was conjectured by Mukai.

+

< < +

Theorem 7.45 ([Wisl]) Let X be a n-dimensional Fano variety of index r(X). If r(X) > dimX + 1 then Pic(X) = Z. 7.3.2 The Nef Value and the Defect Suppose that (X, H) is a smooth polarized variety, i.e. I HI gives an embedding X c JPN. Recall that X* c (iPN)" is the dual variety and def (X, H ) = codim X * - 1 is the defect of (X, H).

Theorem 7.46 If def(X, H)

> 0 then Kx is not nef.

Proof. By Theorem 1.18 X is covered by lines. But it is well-known that if X is covered by rational curves then Kx is not nef. In our case it also follows 0 directly from Theorem 7.10. It follows that, if def(X, H ) is positive, we can apply the technique of the previous section. Theorem 7.47 ([El]) Suppose that X is a nonlinear smooth projective variety in JPN, def X = d > 0, q is a generic point of X and H is generic tangent hyperplane of X at q. Let L = JPd be the contact locus of H with X . Then there exists an irreducible family of lines in X of dimension ( 3 n d - 4)/2. If p is a generic point in X , then there exists a family of lines i n X through p of dimension (n + d - 2)/2.

+


141

Proof. The first statement is a consequence of the standard deformation theory. Indeed, let p E L and let To be a line in L through p. Since

we have n+d-2 3n+d-4 n-d dim H'(TO, NTo,x)= -+ 2 2 2 2

Therefore the Hilbert scheme of lines in X is smooth at the point To and there is a unique irreducible component of the Hilbert scheme containing the point TO.This component has dimension (3n d - 4)/2. The proof of the second 0 statement is similar. It turns out that the nef value and the defect of (X, H) are related:

+

Theorem 7.48 ([BSW, BFSl]) Assume that def(X) > 0. Then (i) The nef value r of X is equal to

r=

dim X

+ def (X, H) + 1.

2 (ii) If SingX n H , H E X*, is a generic contact locus (hence a projective subspace) and 1 E SingX n H is any line, then R = R+[1] is an extremal ray in NE1(X) and c o n t r ~is the nef value morphism cP. In particular, cP contracts Sing X fl H to a point. (iii) If F is a generic fiber of cP then Pic F = Z. (i.1 dim X - def (X, H) dim @(X)5 2 Notice that since dim X E def (X, H) mod 2 by Theorem 7.4, the nef value in this case is an integer. Theorem 7.48 can be used to show that the defect of a polarized variety is equal to 0. One has only to calculate the nef value directly and compare the result with the formula in (i) Proof. We denote n = dim X and k = def(X). *

Step 1 Let P "

Sing X n H , H E X*, be a generic contact locus and let I E Sing X n H

+

be any line. Since (-Kx) . I = - 1 by Theorem 7.10, it follows that 2


142

Step 2 Let @ be the nef value morphism. Since @ is not ample, it follows from the Kleiman Criterion and the Mori Cone Theorem that there exists an extremal rational curve C contracted by @, (Kx +TH) . C = 0. We claim that C . H = 1, i.e. C is a projective line. Indeed, if C H 2 2 then

This contradicts (7.3). Therefore T

= (-Kx)

- C = l(R),

where R = R+ [C]is an extremal ray. Applying Theorem 7.44, wee see that if F is a positive-dimensional fiber of @ then d i m F 27-1.

(7.4)

Step 3

> -+ 1. Let F be a positive dimensional fiber of @, and let 2 x E F. By Theorem 7.13 there exists a line 1 c X passing through x and such

Suppose that

T

+

+

that (-Kx) -1 = le 1. Therefore @ does not contract 1, and hence 1 @ F. +

2

Let y E 1 and y $ F. Arguing as in the proof of Theorem 7.44, we see that there exists an --

+

2

dimensional subvariety Y

c

X covered by rational curves

passing through y that are deformations of 1. Applying a similar argument with ruled surfaces once again, it is easy to see that dim Y n F = 0. However, using (7.4) we get that dimY n

n+k

n+k 2

F 2 dimY + d i m F - d i m X 2 -+ - - n > 0 . 2

This is a contradiction, and (i) is proved. Step 4 Clearly, @ contracts any generic contact locus Sing X n H , H E X*. Moreover, arguing as above (or using Theorem 7.47), we see that any fiber of @ is at least n+k n-k --n + k dimensional, and therefore dimY n - = -. This proves

-

dim F 2

+ 1.

Therefore Pic F = Z by Theorem 7.45. This proves (iii).


143

Step 5 It remains to prove that @ is a contraction of an extremal ray R+ [I]. Suppose, on the contrary, that there exists an extremal ray R such that 1 # R and @ contracts any curve from R. Let C be an extremal rational curve from R. Arguing as in Step 2, we see that 1(R) = (-Kx) C = T. Consider the morphism 9 = c o n t r ~ By . Theorem 7.44, there exists a fiber F of 9 such that dim F 2 T - 1. Let x E F. By Theorem 7.13 there exists a line 1 c X passing through x and such that (-Kx) 1 = T. This line is either a line in a generic contact locus or its limit position, and therefore !P does not contract it. Hence 1 F . Let y E I, y # F . Arguing as in the proof of Theorem 7.44, we see that there exists a (T - 1)-dimensional subvariety Y c X covered by rational curves passing through y that are deformations of 1. Applying a similar argument once again, it is easy to see that dim Y n F = 0. But

This is the required contradiction. 0 Let F be a generic fiber of the nef value morphism. It is a smooth variety with canonical polarization (F, HF), where HF is a hyperplane section of F in the embedding F c X c PN.

Theorem 7.49 ([BFSl]) The following conditions are equivalent: (i) def (X, H ) > 0; (ii) def (F, HF) > dim Y; (iii) def (X, H ) = def(F, HF) - dim Y

> 0.

Proof. It follows from Theorem 7.14 that dim X

+ def (X, H) > dim F + def (F,HF).

Therefore (i) follows from (ii). Obviously (ii) follows from (iii). So it suffices to prove that (iii) follows from (i). Since def (X, H ) > 0 we have T(X,H ) =

dim X

+ def (X, H) + 1 2

by Theorem 7.48. Since F is a generic fiber of @ we deduce that

is a trivial bundle. Therefore KF is not nef and T(F,HF) = T(X,H). If def(F, HF) > 0 then again by Theorem 7.48 def (F, HF) =

dim F

+ def(F, HF) + 1 2

144


and 0 < def (X, H) = - dim X

+ dim F + def (F,HF) = def (F, HF) - dim Y.

It remains to show that if def(X, H ) > 0 then def(F, HF) > 0 as well. Let L = O(H) and LF = O(HF). Recall that for any line bundle L on a smooth variety X we denote by J1(X, L) the vector bundle of 1-jets of L. By Theorem 6.1 we need to show that CdimF(J1(F1 LF)) = 0. From the exact sequence O-+T~(X)@LF -+ J ~ ( X , L ) I F -+ J1(F,LF) -+O we see that the total Chern class

+ +

Since F is a generic fiber of @, T$(X) 63 LF = LF . . . LF (dim Y copies). . In~ particular, ~ Therefore c(T$(X) 63 LF) = (1 H F ) Y ~

+

~dimF(Jl(F9LF))

+

dim Y

dim Y ( . ) C d i m F - i ( h ( ~LF)) l i=l

'

Z

Hk.

All summands on the right are nonnegative since HF is very ample and J1(F, LF) is spanned. Therefore in order to prove that cdim F ( J (F, ~ LF)) = 0 ~ L) IF) = 0. it remains to prove that Cdim F ( J(X, It will be easier to prove the more general statement that ck(J1(X, L)IF) = 0 for k 2 dim F - def(X, H) 1. By the classical representation of a Chern class of a spanned vector bundle [Ful] we see that ck(J1(X1L)) = 0 is represented by a set Ck c X of all points where dimX - Ic 2 generic sections of HO(X,J1(X, L)) are not linearly independent. We may suppose that these sections have a form j l ( ~ l ).,. . ,jl (sdim x-k+2),

+

+

where s l , . . . , Sdimx-k+2 are generic sections of HO(X,L) and jl 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 Ck n F = 0. Since Ck > Ck+l, we only need to prove that C n F = 0,where C = CdimF-def(X,H)+lS~pposethat this intersection is non-empty. Therefore the restriction map Qc : C -+ Y is surjective. Since dim C = dim Y def(X, H) - 1,it follows that a generic fiber of Qc has dimension def(X, H) - 1. Suppose that x E C n F. There exist Xi for 1 i dimY +def(X, H ) - 1, not all zero, such that Xijl (si(x)) = 0. Therefore s = Xisi is a nontrivial global section of L vanishing at x with its 1-jet. Hence the divisor H, is cX singular at x. By Theorem 7.13 there exists a linear subspace containing x, contained in Sing Hsl and such that 6 ( K x+r(X, H)H)lpief(X,H) is contracted by Q is a trivial bundle. Since this is a trivial bundle, to a point and, therefore, c F. On the other hand, since

>

+

+

<
0.

dimX

0. Any weight X E PGIPgives rise to a character of P and therefore to a line bundle LA on GIP. Theorem 7.52

The correspondence a t C, is an isomorphism of QG/pand N?(G/P). The correspondence X + L, is an isomorphism of PGIPand Pic(G/P). The latter group is, in tvrn, isomorphic to N$(G/P). In particular, Pic(G/P) = Z if and only if P is maximal. The pairing PGIPx QGIP+ Z given by duality corresponds to the intersection product on N i (GIP) x N?(G/P). LA is nef if and only if X E P&,. LA is ample if and only if LA is very ample if and only if X E P;&. If X E P : , ~ then the linear system corresponding to LA is base-point-free. The corresponding map given by sections is a quotient G / P -+ G/Q, where Q is a parabolic subgroup such that 17Q is a union of I l p and all simple roots in IIGIPorthogonal to A.

7.4 Flag Varieties with Positive Defect

149

7.4.2 Nef Values of Flag Varieties

Theorem 7.53 ([Sn]) (i) The anticanonical divisor -KGIP corresponds to the weight

KGIPis not nef. (ii) The nef value of an ample line bundle LA given by X

E p;Tp

is equal to

(iii) The nef value morphism @ is a quotient map @ : G / P -t G/Q, where Q is the parabolic subgroup defined by ~IQ = l 7 p U 17', where 17' C nGIP are simple roots for which the above maximum occurs.

Proof. The tangent bundle TGIPis an equivariant vector bundle whose fiber over the identity coset is isomorphic to the quotient of Lie algebras g/p. P acts on it via the adjoint representation on g projected to the quotient. T-weights of this representation are given by A&p. Therefore the anticanonical bundle A ~ corresponds ~ to the weight ~ PGIP. Notice ~ that for ~ any a! E ~ nGIP T

The first product is equal to 2 (it is well-known that CpEA+ = 2 C:=, wi) and the second product is non-positive, because any root from A; is a linear combination of simple roots from l 7 p with non-negative coefficients and the scalar product of two distinct simple roots is non-positive. Therefore P G I P E pfTp and KGIPis not nef. Let

Then, clearly, the nef value T of L is given by T

=

max

aiEno/p

ni ( P G / P ,a ) - max mi

aE%/p

(X,a)

'

Part (iii) follows from Theorem 7.52, part ( 5 ) . Using this theorem, it is straightforward to calculate nef values in particular cases. Suppose that G is a simple group and P is a maximal parabolic = { a t )is a single simple root. Suppose further that subgroup such that Glp L is a generator of Pic G/P, that is, L corresponds to the fundamental weight wi. In the Table 7.4.2 we give dimG/P and the nef value T of L for each G and i.

150


,, i(41fl-3i)/2 i(41-1-3i)/2 l(1 - 1112 16 \

(71

i

Dl

1, ...,1 - 2 1 - 1,l 1.5

Efi

21-i+l 21-2-1 21 - 2 12

Table 7.1.

7.4.3 Flag Varieties of Positive Defect

Now we use these calculations and Theorems 7.48,7.49 to classify all polarized flag varieties with positive defect. Suppose first that G is a simple group and P c G is a maximal parabolic subgroup corresponding to a simple root ai. Then all very ample linear bundles on G I P have the form or k > 0, where L corresponds to the fundamental weight w i . If k > 1then by Theorem 6.6 we have def(G/P, L@" = 0. Therefore it suffices to find def(G/P, L ) . By Theorem 7.48 if def(G/P, L ) > 0 then def (GIP, L) = 27(G/ P, L ) - 2 - dim G I P.


151

Therefore we may look through the table of the previous section and pick all entries such that the RHS d p of this equation is positive. The following table contains all these entries. No.

G

i

r(G/P, L)

dimG/P

dp

1

A1

1,1

1+1

1

1

B4 D5

4 4,5

10 10

4 4

7

A!, 1 odd 2 , l - 1

1+1

21-2

2

8

A5

3

6

9

1

121

G2

1

5

5

3

Table 7.2.

In fact, only the first 6 rows of this table correspond to polarized flag varieties with positive defect, and No. 4 is a particular case of No. 3. This can be verified case-by-case (see [Sn]). The following theorem is an immediate consequence.

Theorem 7.54 ( [ K M ,Sn]) Suppose that Pi is a maximal parabolic subgroup of a simple algebraic group corresponding to the simple root a $ . Let L be a very ample line bundle on G/Pi corresponding to the findamental weight wi. Then (i) def (GIPi, L @ ~=)0 for k

> 1.

152


(ii) def(G/Pi, L) > 0 if and only if G/Pi is one of the following: A1/Pl, A1/R, def(G/P) = d i m ( G / P ) = l . A1/P2, Al/PL-~,1 2 4 even, def(G/P) = 2 , dim(G/P) =2Z-2. 0 Cl/Pl, def(G/P) = dim(G/P) = 21 - 1. 0 B4/P4, D5/P4, D5/B5, def(G/P) = 4, dim(G/P) = 10. Notice that we 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 a simple group.

Theorem 7.55 ([KM, Sn]) Suppose that (GIP, L) is a polarized Bag variety of a simple algebraic group G, where P c G is a non-maximal parabolic subgroup. Then def (G/P, L) = 0. Proof. Suppose that def(G/P, L) > 0. By Theorem 7.53 the nef value morphism is a homogeneous fiber bundle @ : G I P + G/Q, where Q c G is another parabolic subgroup. The fiber F of @ is isomorphic to Q/P. By Theorem 7.48 we have Pic F = Z,and 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 7.49 def(F, L) > dim G/Q > 0. Therefore the polarized flag variety (F, L) is one of the varieties from Theorem 7.54. Let us consider three cases. If F is equal to A1/P2 or All%-1 with 1 4 even, then def(F, L) = 2. Therefore dimG/Q = 1. It follows that G = SL2, Q is a Bore1 subgroup, and there are no possibilities for P . If F is equal to B4/P4, D5/P4, or D5/B5 then def(F, L) = 4. Therefore dimG/Q is equal to 1, 2, or 3. It follows that G is equal to SL2, SL3, SL4, or Sp,. Again none of these groups contains B4 or Dg as a simple component of a Levi subgroup. Finally, let F be A1/Pl, A118,or Cw /PI,and so F is isomorphic to JP1. We want to show that dimG/Q > k, wgich will be a contradiction. The set of simple roots LIQ contains a subset 9 with Dynkin diagram of type Al or C*. Moreover, there exists a simple root a E IT such that a @ 1 7 ~and the Dynkin diagram of 9' = 9 U { a )is connected. Then P'is a set of simple roots of a simple subgroup G' c G and dim G/Q 2 dim G1/Q1, where Q' = G' n Q is a maximal parabolic subgroup defined by 9. It is easy to check that in all 0 arising cases dim G'lQ' > Ic. Now we can handle the case of an arbitrary semisimple group and its arbitrary parabolic subgroup.

>

Theorem 7.56 [KM, Sn] Suppose that (GIP, 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 i n Theorem 7.54 or the following conditions hold: 0

G = G 1 x G 2 , P = PI x

P2,

where PI C G I , P2CG2.


153

L = pry L1 @ pra La, where pri : G / P -+ Gi/Pi are projections and Li are very ample line bundles on Gi/Pi. (G1/Pl, L1) is one of polarized varieties described in Theorem 7.54. d e f ( G ~ / P lL1) , > dimGa/Pz. In this case def (G/P, L) = def (G1/Pl, L1) - dim G2/P2.

Proof. This Theorem easily follows from Theorem 3.5.

0

Remark 7.57 Polarized flag varieties with positive defect were first classified in [KM] using Theorem 3.1. For 'very big' polarizations this result can be deduced from Theorem 6.8. Indeed, the fundamental weights w ~. ,. . ,wk dual to generate the lattice PGIPisomorphic to Pic(G/P). the simple roots in nGIP We shall denote the line bundle L,$ by Li. Then any line bundle on G / P has the form @ ...@ c y - . C=

.qnl

L is very ample if and only if all ni > 0. Then the degree degA(x,L) is a function of nl, . . . ,n,. Let us introduce new non-negative integers mi = ni - 1. Then deg A ( x , ~= ) f (ml, . . . ,m,). All conditions of Theorem 6.8 are clearly satisfied, and therefore f is a non-trivial polynomial with non-negative coefficients. Assume that the weight X satisfies the conditions ni 1 2 (A is very strictly dominant). Then these results clearly imply that the dual variety (G/P, A) * is a hypersurface. Let us consider two examples. Let V = C r , G = SL(V) and let P be the stabiliser of the line in V. Then G / P = P(V). Take X = nwl, where n > 0. Then C1 = O(n). The degree of A(G/p,x)is the degree of the classical discriminant of a homogeneous form of degree n in r variables. Therefore by Example 6.4 we have

In this case f = rmT-l, and so it is a polynomial with nonnegative coefficients. Let G = SL3 and let P = B be the Bore1 subgroup. The weight X can be written as X = nlwl+ n2w2. One can check using the Kleiman formula that

Substituting ml = nl - 1 and m2 = n2 - 1, we see that

Again, this polynomial has nonnegative coefficients.

Dual Varieties of Homogeneous Spaces

Here we study dual varieties and discriminants of several special homogeneous spaces. We start in 8.1 by showing how to use standard results of representation theory such as the Borel-Weyl-Bott theorem, the BGG homomorphism, identities with Schur functors, and formulas of the Schubert calculus to find the codegree of Grassmannians or full and partial flag varieties. We give a list of formulas for the degree of hyperdeterminants and sketch the proof of a Theorem of Zak about varieties of codegree 3. In 8.2 we generalize the Theorem of Matsumura and Monsky about automorphisms of smooth hypersurfaces to automorphisms of smooth very ample divisors on flag varieties. In 8.3 we study commutative algebras without identities from the "discriminantal" point of view. As a corollary we prove that the algebra of diagonal matrices does not have quasiderivations. In 8.4 we study anticommutative algebras (nets of skewsymmetric forms). We show that they have beatiful geometric properties related to cubic surfaces, Del Pezzo surfaces, representation theory of S5, etc. In 8.5 we show that the discriminant in a simple Lie algebra defined by analogy with the disciminant of a linear operator is equal to the discriminant of the minimal orbit, the so-called adjoint variety. Finally, in 8.6 we study related questions about schemes of zeros of irreducible homogeneous vector bundles. In particular, we address a question of classifying irreducible homogeneous vector bundles with a trivial line subbundle, find the maximal dimension of an isotropic subspace of a generic symmetric or skewsymmetric form, and study properties of the related Moore-Penrose involution.

8.1 Calculations of deg X * 8.1.1 Borel-Weyl-Bott Theorem If P is an algebraic group then the exact sequence

156

8 Dual Varieties of Homogeneous Spaces

always splits, i.e. there exists a reductive subgroup L c P that maps isomorphically on P/R,(P). In other words, P is a semi-direct product P = L X &(P), called the Levi decomposition. L is called the Levi subgroup. A subgroup is a Levi subgroup if and only if it is a maximal reductive subgroup of P. All Levi subgroups are conjugated to each other. If G is a reductive group, P c G is a parabolic subgroup and L c P is a Levi subgroup, then we abuse language and call L a Levi subgroup of G. We choose the maximal torus T C L, and then T is also a maximal torus of G. Let B > T be a Borel subgroup such that B n P = B n L. Then B n L is a Borel subgroup in L. We choose the corresponding systems AG > AL of roots, A; > A: of positive roots, and IIG > fi of simple roots in G and L. The semigroup P: of dominant weights of L contains the semigroup of dominant weights of G. For example, if P = B is a Borel subgroup then L = T and PE = X(T). We take any X E Pz, and let Ux be the corresponding irreducible Lmodule. We may consider Ux as P-module with the trivial action of P,. Now we define the fiber bundle G x p Ux as the quotient of G x Ux by the action of P given by: P . (9,z) = (gp-l, P . z).

?Gf

Projection onto the first factor induces the map G x p UA -4 G/P, which is an equivariant vector bundle Ex on G/P. ("Equivariant" means that G acts on Ex compatibly with the action of G on the base G/P.) The Borel-Weyl-Bott theorem shows how to calculate cohomology of Ex. Let Vx be the irreducible G-module with the highest weight X for any X E P&. Let p be the half-sum of positive roots of G. Then (p, a i ) = 1 for any simple root ai E DG.The element z E t i is called singular if there exists a root a E AG such that (z, a) = 0, i.e. if z belongs to the wall of a Weyl chamber. If z is not singular then there exists a unique element w E WG of the Weyl group of G such that w - z belongs to the fixed Weyl chamber C+.

Theorem 8.1 ([Bo])

If X E P: then HO(G/P,Ex) = V,V and Hi(G/P, Ex) = 0 for i > 0. If X p is singular then Hi(G/P, Ex) = 0 for any i . If X + p is not singular, then, for the unique element w E WG,

+

Then ~ ~ ( "(G/P, 1 Ex) = V$ and H i (G/P, Ex) = 0 for i

# 1 (w).

Corollary 8.2 ~ ( E A=) x ( - 1 ) ' d i m H"G/P,

Ex) =

11 (A + p, a)/@, a). a€d&

8.1 Calculations of deg X"

157

Proof. Indeed, if X E Pi, then this is a well-known Weyl formula for the dimension of Vx;see [OV].If X is singular then this formula is obvious. Suppose that X is not singular, and let A' = w(A p) - p E P&for the unique element w E WG.Then

+

dim vAl=

~ ( & x=)

(A'

+ P, a ) / ( ~a) , =

C~EA+,

CYEAZ, The corollary is proved. 8.1.2 Representation Theory of GLn

We choose the standard maximal torus T of diagonal matrices and the Bore1 subgroup of upper-triangular matrices. The group of characters X(T) is freely generated by characters ~ 1 ,. .. ,E,, where

The roots are aij = Ei - E j . The root subspace of aij is CEij. The simple is dominant if and only roots are ~1 - E Z , . . . ,~ ~ -- En.1 A weight X = if X1 2 Xz 2 . . . 2 A,. For example, the highest weight of SdCn is d~~ and the corresponding highest weight vector is ef. Similarly, the highest weight of AdCn is & I + . . . + ~ dand the corresponding highest weight vector is el A.. .Aed. Onedimensional representations of GL, correspond to characters of GL, , i.e. to functions detP, p E Z. The corresponding highest weights are p E i . Therefore any irreducible representation, after tensoring by a suitable character, has the highest weight X with the property A, 2 0. We identify these dominant weights with the set P, of all partitions

x

x

Let

X

E

P?

C P n be the subset of partitions such that X1 5 m. For any partition Pn we set

For any partition X E given by

1x1 = XI

Pp

+ . . . + A,.

we denote by At E

Ph

the transposed partition

A: = max{j 1 Xj 2 i).

For any partition X E P, (or just the dominant weight A), we denote by Sx(Cn) the irreducible representation of GL, with the highest weight A.

158


We will need the Cauchy formula

the decomposition of GL, x GL,-modules. A parabolic subgroup P c GL, consists of block triangular matrices

and its Levi subgroup L consists of block diagonal matrices with Aij = 0 for and C ni = n. Dominant weights X for L are given i > j . Here Aij E by sequences XI, . . . ,An such that

By Corollary 8.2, the Euler characteristic x(£A) is given by the formula

8.1.3 Dual Variety of the Grassmannian Let X = Gr(n1, be the Grassmannian of nl-dimensional subspaces in Cn1+,2 in the Pliicker embedding X c IP(CN), where CN = AnlCn1fn2. We are going to calculate degree of the dual variety X*.

Theorem 8.3 ([Las]) Let X = Gr(nl, Cnlfn2) C P(Anl@nl+nz)be the Grassmannian of nl-dimensional subspaces in Cnl+n2 in the Plucker embedding. Then

where S = nlnz - IXI or S = nln2 - IX( - 1, p E Z is arbitrary, and

Proof. Let C = O x ( l ) , and we have dimX = nlnz. By Theorem 6.13,

for any p E Z. We have

8.1 Calculations of deg X*

A"[Ox] - 2[LV]) = (-l)"(lc

159

+ 1)[LVIk- k[LV]"').

Let S be the tautological vector bundle on X, and let V / S be the quotient tautological bundle. Then T: = S 8 (V/S)' and by the Cauchy formula,

Therefore we have

where S = nln2 - IAI or S = nlnz - (XI - 1 and p E Z is arbitrary. By Corollary 8.2,

t v(-Ak7 -Am-17 . . , - A ~ , A ~ - P +,..., S An-p+S)

V ( 1 , .. .,nl +n2) The theorem is proved. 0 If m = 2 then this degree was already calculated in Example 2.13. For small values of m and n the degree can be found from the following table.

Table 8.1.

8.1.4 Codegree of G / B Let G be a simple algebraic group of rank r with a Bore1 subgroup B. The minimal equivariant projective embedding of G / B corresponds to the line bundle Lp, where p=w1+

...+w,

is a half sum of positive roots. We shall follow [CW] here and calculate the degree of the corresponding discriminant. Let {Dl, . . . ,DN)= A+ be the collection of all positive roots. We consider the matrix

160


Let P s .( M.) be the sum of all permanents of s x s submatrices of M. Recall n that the permanent of a n x n matrix (aij) is equal to

C

nES,

n ai,g(i)

i=1

Theorem 8.4 ([CW])

Proof. Recall that by Theorem 6.2 we have

C(i+ l)]GIB cn--i(T;//g)

deg d ( G l ~ ,=L , )

.

i=O

Therefore we need to calculate these integrals. The correspondence A t - i LA identifies P 8 Q with Pic(G/B) 8 Q. Therefore we have a homomorphism of commutative algebras

where S o ( P @ Q) is the symmetric algebra of P 8 Q and Ae(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 of positive degree (Se(P@ Q ) ) y , where W is the Weyl group. For example,

Using an appropriate filtration of the cotangent bundle T&B and 'the splitting principle' it is easy to see that

Therefore it suffices to compute c(piXi). For any root a we define a linear function

It extends by a Leibniz rule to the differential operator on Se(P@Q).Consider the differential operator D given by


161

D decreases the degree by N, and so in particular we get a linear form

@ Q) we have One can show (see [CW]) that for any x E sN(p

Now the claim of the theorem follows by an easy calculation.

0

8.1.5 A Closed Formula In most circumstances, known formulas for the degree of the discriminant depend on a certain set of discrete parameters. There are two possibilities for these parameters. First, we may fix a projective variety and vary its polarizations, see Theorem 6.8. The second possibility is to change a variety and to fix a polarization (in a certain sense). For example, consider the irreducible representation of SL,, with the highest weight A. Then this weight can be no with respect to the considered as a highest weight of SL, for any n natural embedding SL, c SLn+l C . . .. As a result, we 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, see e.g. Theorem 8.3. However, sometimes this function has a closed expression - the Boole formula (6.4) is a formula of this kind. Another simple example is the formula (2.13) for the degree of the dual variety to Gr(2, n) in the Pliicker embedding. The following theorem is a mixture of these two cases.

>

Theorem 8.5 ([TI]) Let V be an irreducible SL,-module with the highest weight ( a - l ) p l + p2, where a > 2. Then the variety X* c P ( V V ) projectively dual t o the projectivization X ofthe orbit ofthe highest vector is a hypersurface of degree (n2 - n)an+l - (n2 n)an-l - 2n(-l)n

+

(a

+

Proof. We have to calculate the degree of Ax.We use Kleiman's formula (6.2) for the degree of the dual variety. In our case X = G I P , where G = S L , and P c G is the parabolic subgroup of matrices


162

Assume that T c G is the diagonal torus, B is the Bore1 subgroup of lowertriangular matrices, X I , . . . ,x, 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 II S, is the Weyl group of G, and W p N Sn-2 is the Weyl group of P. It is well known (see [BGG]) that the map c : X(T) -+ Pic(G/B) that assigns to X the first Chern class of the invertible sheaf LAcan be extended to a surjective homomorphism c : S 4 Am(G/B)in the (rational) Chow ring, and its kernel coincides with SYS. The projection a : G I B + G / P induces an embedding a* : A' (GIP) t A' (GIB). The image coincides with the subW pdenote . the algebra of Wpinvariants. Hence A0(G/P) = S W p / ~ ~ S We homomorphism SWpt Am(G/P)by the same letter c. To apply Kleiman's formula we need cl ( L ) (which is equal to c(axl+ x2)) and the total Chern class of T,: which is equal to

(This can be shown using a filtration of Tg and the splitting principle.) Let al, . . . ,a,-2 be the elementary symmetric polynomials in 23,. . . ,x,. Then SWp= Q[xl,2 2 , a1, . . . ,a,-2],and the ideal sYSWpis generated by

+

+ +

Sy

SWp , and A' (G/P) is Hence ai = (- l)i(xi 22-I 2 2 . . . xi) mod isomorphic to the quotient ring Q[xl, x2]/(fl,f2), where fl = X ; - ~ + X ; - ~ X ~ . . . x;-l and f2 = xy . Note that fl ,f2 is the Grobner basis of the ideal (fl, f2) with respect to the ordering x2 > x1 (see [Berg]). Hence the set of XiYj is a basis of the quotient algebra, where X = xl mod (fl, f2) and Y =x2 mod ( f 1 , f 2 ) , i = 1,...,n - l , j = l , ..., n - 2 . To calculate the degree of the discriminant by Kleiman's formula, we have - ~Let Go be the longest element in W, and let wo to calculate SZ c ( x ~ - ' Y ~ ). be the shortest element in GoWp with the reduced factorization

+

Let

be the corresponding endomorphism of S of degree -(2n - 3), where

+


and s(ij)is the reflection that transposes xi and

L

Xj.

163

Then

C ( X ~ - ' Y ~ -= ~ )A,, (x:-~x;-~)

(see [BGG]). It is obvious that

wherepk = (n-1, n)(n-2, n-1) . . . (k, k+l). These factors are both equal to 1, since they are equal to J&, el (O(l))n-l and &,n-2 c1(O(l))n-2, respectively. We finally see that J, c(Xn-'Yn-') = 1. It remains to calculate the polynomial

in the ring Q[X, Y] (with the basis xiyj,i = 1, . . . ,n- 1,j = 1,. . . ,n -2, and relations Xn-l Xn-2Y . . . Yn-l = 0, Yn = 0 and X n = 0, which follow from the preceding relations), where ck is the kth homogeneous component of the polynomial

+

+ +

in which the ith symmetric function of 2 3 , . . . ,xn must be replaced by Xi-lY . . . Yi). By the previous discussion, the result of this (-l)i(Xi calculation will be deg(Ax)Xn-l Yn-2. Note that the polynomial (8.2) is equal to

+

+ +

where

We have, further, F3(T) =

r2

x i + l - yi+l C ( T -~ ) ~ - ' - ~ ( - l ) ~ i=O X-Y

164


X-Y

We obtain, likewise, that

F4(T)=

- yi+l C(T- ~ ) " - ~ - ~ ( xi+l -l)"

i=O ("-'

X-Y

X-Y

We deduce from the latter formula that

F 1 ( T )=

n ( X - Y ) ( T - X)n-l + ( - X ) n - ( - Y ) n x X-Y

Using the elementary formulas

=

8.1 Calculations of deg X *

we obtain

Putting T = ax

where

+ bY in the latter formula, we obtain

165

166


It remains to calculate the degree of the discriminant, that is, the difference ~ -Xn-'Yn-' ~ in the expression for between the coefficients of x ~ - ~ Yand F t ( a X by). After some transformations we obtain

+

This is the degree of the discriminant of the irreducible SLn-module with the highest weight (a - b)cpl + bcpz. Substituting b = 1 in the last formula, we obtain

8.1.6 Degree of Hyperdeterminants Consider the flag variety P" x . . .x Pkr of the group SLkl+l x . . .x SLk,+l. The projectively dual variety of its 'minimal' equivariant projective embedding is related to a nice theory of hyperdeterminants initiated by Cayley and Schlafli [Cal, Ca2, Schl]. Let r 2 2 be an integer, and let A = (ail,..ir), 0 5 ij 5 kj be an rdimensional complex matrix of format (kl 1) x . . . x (& 1). The hyperdeterminant of A is defined as follows; cf. Example 3.8. Consider the product X = Pkl x . . . x Pkr of several projective spaces embedded in P(kl+l)X...X(k~+l)-l via the Segre embedding. Let X * be the projectively dual variety. If X * is a hypersurface then it is defined by a corresponding discriminant A x ,which in this case is called the hyperdeterminant Det. Clearly Det(A) is a polynomial function in matrix entries of A invariant under the action of SLkl+' x . . . x SLk,+'. If X* is not a hypersurface then we set Det = 1.

+

+


167

Example 3.8 shows that X * is a hypersurface (and, hence, defines a hyperdeterminant) if and only if 2k.j kl . . . k, for j = 1,.. . ,r. If for one of the j we have an equality 2k.j = kl . . . k, then the format is called boundary. Let N(kl, . . . ,k,) be the degree of the hyperdeterminant of format (kl 1) x . . . x (k, 1).The proof of the following theorem can be found in [GKZZ] or [GKZ3].

< + + + +

+

+

Theorem 8.6 (i) The generating function for N(k1, . . . ,k,) is given by

where ei(zl, . . . ,z,) is the i-th elementary symmetric polynomial. (ii) The degree N(k1,. . . ,k,) of the boundary format is given by (assuming that kl = k2 . . . k,)

+ +

N(k2

+ .. . + k,, k2,. . . ,k,)

=

(52

+. . - + kr + I)! kz!. . . k,!

(iii) The degree of the hyperdeterminant of the cubic format is given by

(iv) The exponential generating function for the degree N, of the hyperdeterminant of format 2 x 2 x . . . x 2 (r times) is given by

8.1.7 Varieties of Small Codegree Let X be a projective variety. We define the codegree codegX by codeg X = deg X*. If X* is a hypersurface, then codegX is also called the class of X . This is a classical invariant playing an important role in enumerative geometry. If X * is not a hypersurface and X' is a generic hyperplane section of X, then by Theorem 5.3 it is clear that codeg X' = codeg X. With regard to codegree, the most simple nonsingular projective varieties are those whose codegree is small. The problem of classification of varieties of small codegree should be compared with that of classification of varieties of small degree. Much is known about this last problem. The case of varieties

168


of degree two is classical (quadrics). The complete description of varieties of degree three was given by A. Weil in the anonymous publication [XXX]. Swinnerton-Dyer [Sw] classified all varieties of degree four. In the smooth case the classification up to degree 8 has been completed in several papers by Okonek [Ol, 02, 0 3 , 041 and Ionescu [I, 11, 121. Fania and Livorni [FaLl, FaL21 classified varieties of degree 9, 10. The same question can be asked for the codegree, but this problem is quite different. For example, in the case of varieties of small degree one can proceed by induction using the fact that the degree is stable with respect to passing to hyperplane sections, whereas there is no such inductive procedure for codegree.

Theorem 8.7 ([Z6]) There exist exactly 10 non-degenerate nonsingular complex projective varieties of codegree three, namely 0 0

the self-dual Segre threefold P1 x P2 c P5, its hyperplane section Fl C P4 obtained by blowing up a point i n P2 by means of the map defined by the linear system of conics passing through this point, the four Severi varieties, i.e. the Veronese surface vz(P2) c P5, the Segre variety P2 x P2 c P8, the Grassmann variety Gr2(C6) c PI4 of lines in P5 and the 16-dimensional variety E C PZ6 corresponding to the orbit of highest weight vector for the standard representation of the algebraic group of type E67 the four varieties obtained by projecting the Severi varieties from generic points of their ambient linear spaces.

Sketch of the proof. If X* is not a hypersurface, then we can either apply Weil's classification theorem to X * or consider the intersection of X with a general hyperplane, thus reducing the problem to the case when X * is a (singular) cubic in ( P ~ ) Suppose ~ . that X* is a hypersurface, and let C = X;,. Then each point of E has multiplicity two and, since degX* = 3, we X*. Let prl and pr2 denote the projections of the conclude that Sec(C) conormal variety Ix c PN x (PN)Von X and X*. Let x be a generic point of X , let P, = pr2(pr,1(x)), and let

c

Ex = {H E P, I x is not a non-degenerate quadratic singularity of H n X ) It is easy to see that C, c CnP, is either a hyperplane or a quadric in P,. In the first case one can show that C is a linear subspace in (PN)Vfrom which it is possible to deduce that X = F I . In the second case Sec(E) = X * and one can show that C is nonsingular and, either dim C = n and X is a Severi variety, or dim C = n - 1and X is a nonsingular projection of a Severi variety 0 from a point. The classification of smooth varieties of codegree 4 is still unknown. All known examples arise from Pyasetskii pairing (2.2). The conjectural list consists of

8.2 Matsumura-Monsky Theorem

0

169

The Segre embedding of P1x Q, where Q is a quadric hypersurface. The twisted cubic curve v3 (P1). The Pliicker embedding of the Grassmannian Gr3(C6). The isotropic Grassmannian ~ r g ( C of ~ )isotropic 3 dimensional subspaces in the symplectic space C3 in its minimal equivariant embedding. The spinor variety 86. 27-dimensional variety E c P55 corresponding to the orbit of highest weight vector for the minimal representation of the algebraic group of type E7. The Segre embedding of P1x P3 (self-dual variety with defect 2), its nonsingular hyperplane section and its non-singular section by two hyperplanes (which is either P1 x Q1 or the blow-up of a projective quadratic cone in its vertex).

8.2 Matsumura-Monsky Theorem Let D c Pn 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 c G / P be a smooth ample divisor. Let LA = O(D) be the corresponding ample line bundle, where X E P+ is a dominant weight. Then one might expect that the normalizer NG(D) of D in G is finite if X is big enough. Indeed, if NG(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 X 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 A. Much better estimates can be obtained generalizing the original proof of the Matsumura-Monsky theorem. The problem can be reformulated as follows. Suppose that Vx is an irreducible G-module with the highest weight A. Let V c Vx be the discriminant variety (the dual variety to the orbit of the highest weight vector in V):. We will show that if X is big enough then any point x E Vx \ V has a finite stabilizer G, and the orbit of x is closed, Gx = (therefore x is a stable point of Vx 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 [Ell] and [El21 to handle irreducible representations of semisimple groups and any representations of simple groups).

170


If V is not a hypersurface then an easy inspection using Theorem 7.56 shows that the stabilizer of any point is infinite. So from now on we shall assume that V is a hypersurface defined by vanishing of the discriminant A. A dominant weight X is called self-dual if Vx is isomorphic to V : as a G-module. Let P i c P+ be the subcone of self-dual dominant weights. Let y be the highest root.

Theorem 8.8 Let Vj, be a n irreducible representation of a simple algebraic group G with the highest weight X such that V i s a hypersurface. Suppose that (A-y,p) > 0 for any p E P;. Let x E VA\V. Then G, is finite and Gx = G. Moreover, GlXIi s also finite, where [XI is the line spanned by x. Proof. If G, is finite and G[,] is infinite then [x]\{O} C Gx. Therefore 0 E Gz and A(x) = A(0) = 0, and hence, x E V. The same argument shows that if x E VA\ V then ?%c Vx \ V. If Gx is not closed then G, is infinite for any point y E ?%\ Gx. Therefore, in order to prove the theorem, it suffices to prove that G, is finite for any x E VA\ V. Suppose that G, is infinite. Then G, is infinite and reductive (see e.g. [PV]) for any point y from the closed orbit in G.Therefore, it suffices to prove that if S c G is a one-dimensional torus and Sx = x then x E V. Without loss of generality we may assume that S c T, where T is the fixed maximal torus, and t = LieT is the Cartan subalgebra. Let x = C,,? x, be the weight decomposition of x. Let Supp(x) = { T E P I x, # 0) be the support of x. For any p E P let H, denote the hyperplane of weights perpendicular to p. Then there exists p E P such that Supp(x) c H,. Using the action of the Weyl group we may assume that p E P + . For any X 6 P+ let XV be the highest weight of the dual module V c Then XV = -wo(X), where wo is the longest element of the Weyl group. Suppose that (p, XV -7) > 0. Then for any positive root a we have (p, XV a ) > 0. Therefore x is perpendicular to [g,vxv 1, where vxv is the highest weight : It follows that x E V. vector of .V Suppose that (pV,XV - y) > 0. Then for any positive root a we have (pV,XV - a ) > 0. Therefore wo(x) is perpendicular to [g,vxv]. It follows that wo(x) E 'D and hence x E V. Suppose now that (p, XV - y) 5 0 and (pV,XV - y) I 0. Then

+

But p pV is a self-dual weight, and hence this contradicts assumptions of 0 the theorem.

Example 8.9 If G = SL, and X = C niwi, where w l , . . . ,w, are the fundamental weights, then the conditions of the theorem are satisfied if and only if C ni > 2; for example if X = nwl and n > 2. In particular, we recover the original Matsumura-Monsky Theorem.

8.3 Discriminants of Commutative Algebras

171

8.3 Discriminants of Commutative Algebras 8.3.1 Commutative Algebras Without Identities Let V = en.Consider the vector space A = S2VV@ V parametrizing bilinear commutative multiplications in V. In the sequel we identify points of A with the corresponding commutative algebras.

Definition 8.10 Let A E A. A non-zero element v E A is called a quadratic nilpotent if v2 = 0. Let Dl c A be the subset of all algebras that contain quadratic nilpotents. A one-dimensional subalgebra U C A is called singular if there exists linear independent vectors u E U and v E A such that u2 = a u ,

a uv = -v, 2

where a E 6.

Let 'D2 c A be the subset of all algebras that contain singular subalgebras. Then the following theorem holds.

Theorem 8.11 ([T6]) (i) 'Dl and D2 are irreducible hypersurfaces. (ii) Let A E A. Then A contains a one-dimensional subalgebra. (iii) Let A E A \ (Dl U D2). Then A contains exactly 2n - 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 E VV defines a homogeneous quadratic function

Then 'Dl corresponds to the set of linear systems of quadrics with zero resultant. This proves that Dl is an irreducible hypersurface. There exists, however, another useful identification. Any algebra A determines the n-dimensional linear system of a f i n e quadrics in V. Namely, any linear form f E VV defines an affine quadratic form

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.The base points of this linear system that do not lie on the infinite hyperplane 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 (iii) follows from the Bezout theorem.

172


D2 is irreducible since D2 = GL, .DL, where DL c 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 D2 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. Algebras that do not belong to discriminant varieties Dl and D2 are called regular. Both hypersurfaces Dl and D2 can be interpreted as standard discriminants. First, we can enlarge the symmetry group and consider S2((Cln)v@Cn as an SL, x SL,-module. Then this module is irreducible and its discriminant variety (the dual variety of the projectivization of the highest weight vector orbit) coincides with Dl. Now, consider A = S2VV8 V as an SL(V)-module. d, where dois a set of algebras Then this module is reducible, A = A. with zero trace and d is isomorphic to VV as an SL(V)-module. Consider the discriminant of as a function on A (forgetting other coordinates). Then the corresponding hypersurface is exactly D2. If we consider the set of linear operators Hom(V, V) = VV @ V instead of A, then these constructions give rise to the determinant and the discriminant of a linear operator (see Example 2.11, Theorem 8.25).

+

8.3.2 Quasiderivations

Let g be a Lie algebra with representation

Consider any v E V. A subalgebra

is called the annihilator of v. The subset

is called the quasi-annihilator of v. Clearly, g, c Qgv. Of course, the quasiannihilator is not a linear subspace in general, However, we have the following version of a Jordan decomposition:

Lemma 8.12 Suppose that p is the differential of the representation of an algebraic group. Let g E Qgv. Consider the Jordan decomposition in g, g = g, g,, where g, is semisimple and g, is nilpotent, and [g,, g,] = 0. Then gs E 0, a n d g n E Qgv.

+

The proof is easy. Now we apply this construction for g = gI, and p being the natural representation in the vector space VV@VV@Vthat parametrizes algebras (bilinear

8.3 Discriminants of Commutative Algebras

173

multiplications in V). Let A be any algebra. Then g~ is identified with the Lie algebra of derivations Der(A). Operators D E QgA are called quasiderivations. Of course, it is possible to write down explicit equations that determine Q Der(A) = QgA in End(A), but this formula is quite useless (see [Vil]). We shall use a particular case of it that is quite easy to verify:

Lemma 8.13 Let A be an algebra, D E End(A), where D2 = 0. Then D E QDer(A) if and only if

for any x, y E A

.

Example 8.14 It was conjectured in [Vil] that all quasiderivations of the algebra of matrices Mat, have the form D(x) = ax+xb, where (a+ b)2 = [a,b]. Let us give a counterexample. Consider the linear operator D(x) = exe, where e2 = 0, e # 0. Then it is easy to check using (8.3) 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 AD with the same underlying vector space but with the new multiplication

Then, if A satisfies a polynomial identity, AD satisfies this identity as well. More generally, let p : g -+ End(V) be the differential of a representation of an algebraic group G, and let v E V and D E Qgv. Suppose that H C V is a closed conical G-equivariant hypersurface and v E H. Then p(D)v also belongs to H. Indeed, since H is equivariant, exp(Xp(D))v belongs to H for any X E C. Since D is a quasiderivation, exp(Xp(D))v = v Xp(D)v. Since H is conical, v/X p(D)v belongs to H. Since H is closed, p(D)v also belongs to H. Now suppose that A is a regular commutative algebra. Then we claim that Q Der(A) = 0.

+

+

Theorem 8.15 ([T6]) Let A E A and A $DlU D z . Then QDer(A) = 0. Proof. By Lemma 8.12 it suffices to check that A has no semisimple derivations and no nilpotent quasiderivations. Suppose that Der(A) contains a nonzero semisimple element. Then the group Aut(A) contains a one-dimensional algebraic torus T. Let t be its Lie algebra, H E t, and H # 0. We way assume that the spectrum of H in A is integer-valued. Let A, c A be a weight space of weight n. Then A = @A, is a Z-grading. Let v E A be a homogeneous element of a maximal positive (or minimal negative) degree. Then v2 = 0, and hence A E Dl.

174


Suppose now that E is a non-zero nilpotent quasiderivation. We can embed E in an sl2-triple (F,H, E) c End(A). Let

be an s12-moduledecomposition, where Rd is an irreducible (d+l)-dimensional module. Consider also the weight decomposition A = $An

and A = $An

with respect to H. Let

Jn = An + ~

+

, + l

+ .. ..

To avoid the abuse of notations, denote by a E A the point corresponding to the algebra A. Let Supp a be the support of a (i.e. all n E Z such that a, # 0, where a = Can,a, E An). Since E2a = 0, the vector Ea is a linear combination of highest weight vectors, and therefore Supp a c {-1,0,1,2, . . .). This is equivalent to JnJmC Jn+m-l. In particular, if v E A is a weight vector of the weight n then v2 = 0 since n > 1. Since A is regular, it follows that - ' $ A~ O $ A~ '. ~ ~ = r n ~ & $ = Am Since A1 = J1,A' is a subalgebra in A. By Theorem 8.11 (ii), A1 has a one-dimensional subalgebra U spanned by an idempotent u (since A has no quadratic nilpotents). Let v E A-l be the unique vector such that Ev = u. Since E2 = 0, we can apply formula (8.3) with x = y = v. We get E ( v ) ~= 2E(E(v)v), and hence u = 2E(uv). Therefore uv - i v E JO.Notice that the operator of left multiplication by u preserves J O .Since this operator has an eigenvector in A/JO with eigenvector 112, it has such an eigenvector in A. Therefore U is a singular subalgebra in A. 0 The following Corollary was proved in [An] using other methods.

Corollary 8.16 ([An]) Let A be an n-dimensional semisimple commutative algebra, i.e. a direct sum of n copies of @, i.e. the algebra of diagonal n x n matrices. Then A has no nonzero quasiderivations. Proof. It is sufficient to check that A # V1 U V2. Clearly, A has no nilpotents. Suppose that U C A is a one-dimensional subalgebra spanned by an idempotent e E U . Since the spectrum of the operator of left multiplication by e consists of 0 and 1, it follows that U is not singular. 0

8.4 Discriminants of Ant icommutative Algebras 8.4.1 Generic Anticommutative Algebras

If the set of certain objects is parametrized by an algebraic variety X then we can speak about generic objects. Namely, we say that a generic object

-

8.4 Discriminants of Anticommutative Algebras

175

satisfies some property if there exists a dense Zariski-open subset Xo c X such that objects parametrized by points of Xo share this property. Sometimes it is possible to find a discriminant-type closed subvariety Y c X and to study properties of 'regular' objects parametrized by points from X \ Y. For example, instead of studying generic hypersurfaces it is beneficial to study smooth hypersurfaces. In this section we implement this program for quite non-geometric objects, namely anticommutative algebras. Let V = Cn. We fix an integer k, 1 < k < n - 1. Let d n , k = AkvV€4 V be the vector space of k-linear anticommutative maps from V to V. We identify points of Antkwith the corresponding algebras, that is, we assume that A E d 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 8.17 ([TI]) Let A E An,k be a generic algebra. Then (i) Every m-dimensional subspace is a subalgebra i f m < k. (ii) A contains n o m-dimensional subalgebras with k 1 < m < n. (iii) The set of k-dimensional subalgebras is a smooth irreducible (k- l ) ( n - k)dimensional subvariety i n the Grassmannian Gr(k, A). (iv) There are finitely many ( k + 1)-dimensional subalgebras, and their number is

+

+ +

where 1A1 = XI . . . X k + l , 1111 = p1+ . . . f pk+l, 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

+

+

Proof. The GLn-module An.kis a sum of two irreducible submodules:

Here is isomorphic to A"'V~: we assign to every ( k - 1)-form w the algebra with multiplication

176


Note that every subspace of this algebra is a subalgebra. Hence the lattice coincides with the lattice of subalgebras of A', of subalgebras of A E where A H A0 is the GLn-equivariant projector on the first summand in (8.4). Algebras in will be called zero trace algebras, since A E A:,k if and only if the (k - 1)-form Tr[vl, . . . ,vk-1, .] is equal to zero. Hence the theorem will be proved once we have proved it for generic algebras in A:,k. We choose a basis {el,. . . ,en) in V, identify GL, 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 L k. Consider the parabolic subgroup of matrices

where B is an m x m matrix and A is an (n - m) x (n - m) matrix. Then G / P coincides with Gr(m, V). Consider the vector bundle

on G I P , where S is the tautological bundle and V/S is the factor-tautological bundle. Then the assumptions of Theorem 8.27 are fulfilled since C = LA, where X is the highest weight of A:,k. Therefore

Let A E A:,k, and let s~ be the corresponding global section. Then the scheme coincides with the variety of m-dimensional subalgebras of A. of zeros (Zs,)red Let us return to the theorem. Statement (i) is obvious. (ii) follows from Theorem 8.27 (i). Theorem 8.27 (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 - l)(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 @v@ 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 = A%" 8 V/S over Gr(k, V) corresponding to A has transversal intersection with the zero section. Then the Koszul complex

+

is exact by Theorem 5.17. Note that ApLV is isomorphic to the bundle S ~ A ~8SAP(V/S)". This is a homogeneous bundle over G / P of the form L,, where

8.4 Discriminants of Anticommutative Algebras

177

and ~i are the weights of the diagonal torus in the tautological representation. Note that the weight p + p (where p is the half-sum of the positive roots) is singular (belongs to the wall of the Weyl chamber) for any p, 1 p n- k. By the Borel-Weyl-Bott Theorem 8.1, H0(Gr(k, V), ApCV) = 0 for 1 p n- k. Hence HO(Z(s),Oz(,)) = HO(Gr(k,V), 0 ) = C, as was to be shown. We postpone the proof of (v) till the end of this section and consider (iv) and (vi), that is, we shall calculate the highest Chern class of the bundle AkSv @ V/S over Gr(k 1,V). We use the standard notation, facts, and formulae from the Schubert calculus (cf. [Full. The letters X and p 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 X is denoted by ox, where ox E AI'I ( ~ r ( k 1,V)). (We grade the Chow ring by the codimension, (XI = X1 . . . X k + l , where Xi is the length of the ith row of X.) We need the total Chern class of the bundles S and V/S:

< < <

+

(d+:-2) + 1. Lemma 8.36 Let d 2 3 and a _> 2. Then (d+a-l) > -a-1 It follows from Lemma 8.36 that

B. Skew-symmetric Case. Clearly if i > k - d then the conditions of Proposition 8.35 are satisfied. Therefore it suffices to check when the inequality

n2follows from the inequality

k-i

(3

n>-+k k as i = I, . . . ,min(k - d, n - 5). If d = 1 then Theorem 8.33 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:

>

8.6 Homogeneous Vector Bundles

195

It follows from Lemma 8.37 that

It remains to consider 3 cases: i = k - d - 2, i = k - d - 1, i = k - d as iln-k. Let i = k - d - 2, n 2k - d - 2. We want to deduce the inequality

(dt62) d+2 + 2k

>

> $+ +

2 = 2k - 2 from n k. I t suffices to check 2 k(k - 2 ) . Since d 3 and k - d = i 2 2 3 we have (;) L (;) L k(k - 3) > k(k - 2). 2k - d - 1. We want to deduce the inequality Let i = k - d - 1, n

(i)

-d -

9

>

9

>

,+, +

2k - d - 1 = 2k - d. n > $$+2k-d-1=2k-dfromn> k(k - d). Since d 3 and k - d = i 1 2 2 we It suffices to check L 2 k(k - 3) 2 k(k - d ) as k - d 3. If k - d = 2 then have 2 2(d 2) k(k - d). = (d+2)(d+1) 2

(5)

(2

(5) >

(t)

>

+ >

Let i = k - d, n

2 2k - d. We need

> ++

+

>

to deduce the inequality n

> (3 +

("1 k. All exceptions are defined by the 2k - d = 2k - d from n following system of equalities and inequalities:

+

+

+

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 E AnP2VV has a (n - 1)-dimensional isotropic subspace or not. This holds iff a generic 2-form in VV 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 E A3(C7)' has a 5-dimensional isotropic subspace. In suitable coordinates this isotropic subspace will coincide with the linear span (el,. . . , e 5 ) . Therefore w = aijkXiAXjA~k,~hereaijk=Oask

Theorem 8.44 0 is a Moore-Penrose orbit if and only if ht(O) = 2. In this case for any e E 0 there exists a unique s12-triple (elh, f ) such that h E i t . Proof. Let x + Z denotes a complex conjugation in g with respect to the compact form t. Therefore x = Z iff x E t and x = -Z iff x E it. Let B ( x ,y ) = Tr ad(x)ad(y) be the Killing form of g. Finally, let H ( x ,y) = - B ( x , p) be a positive-definite Hermitian form on g. Lemma 8.45 We jix a nilpotent element e E g. Suppose that (elh, f ) is an sI2-triple in g such that h E it. Then for any other sI2-triple (elh', f ' ) we have H(h, h ) < H(ht,h'). In particular, if there exists an sI2-triple (elh, f ) with h E it then the &-triple with this property is unique. Proof. Recall that if (elh, f ) is an s12-triple then h is called a characteristic of e. Consider the subset 'H c 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 a:(e) of the centralizer aB(e) in g of the element e. Since H ( h t ,h') is a strongly convex function on 31, there exists a unique element ho E 7-1 such that H(ho,ho) < H(hl,h') for any h' E 'H, h' # ho. We need to show that ho = h. It is clear that an element ho E 'H minimizes H ( h ,h) on 3-1 iff H(ho,a: ( e ) ) = 0 iff B(&, 2: (e)) = 0. If h E 'H n ito then = -h and we have

x

0 Therefore h = ho. Suppose that (elh, f ) is an s12-triple in g. Consider the grading g = $ gk k

such that x E gk iff [h,x] = kx. Let n+ = known that a:(e)

c n+.

Lemma 8.46 Suppose that Penrose orbit.

a:(.)

$ k>O

gk: and n- =

@ kn

the contrary, that at(e) # n+. Let J E g,, p > 0, be a homogeneous element that does not belong to 3; (e). Let u = exp(6) and e' = Ad(u)e. We claim that all characteristics of e' do not belong to iK. Indeed, all characteristics of e' have the form Ad(u)h+Ad(u)x, where x E a;(e). Suppose that for some x we have Ad(u)h+ Ad(u)x E iK. Since h E iK, fi* = nT, and Ad(u)(h x) - h E n+ it follows that Ad(u)(h x) = h. In n+ modulo @ gk we obtain the equation

+

+

k>p

+

[J, h] x = 0, but [h,J] = pJ and therefore J E $(e). This is the required 17 contradiction. Now we can finish the proof of Theorem 8.44. Combining previous lemmas we see that 8 is a Moore-Penrose orbit if and only if ai(e) = n+. It follows from the .&-theory that dimat(e) = dim g1 dim g2. Therefore 3; (e) = n+ if and only if g, = 0 for p > 2. Clearly, this is precisely equivalent to ht(O) = 2. In this case for any e E 0 there exists a unique s12-triple (e, h, f ) such that 0 h E it by Lemma 8.45. Now let us turn to the graded situation. Suppose that g is a %graded simple Lie algebra with g = @ gk. Let P c G be a parabolic subgroup with

+

kEZ

Lie algebra p =

$ gk. kro

Let L

c P be a Levi subgroup with Lie algebra go. We

choose a compac 0 and any L-orbit O c gk. Then O is called a Moore-Penrose orbit if for any e E O there exists a homogeneous &-triple (e, h, f ) such that h E ito. In this case f is called an MP-inverse of e. A grading is called a Moore-Penrose grading in degree k > 0 if all L-orbits in gk 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 C G is called a Moore-Pen,rose parabolic subgroup if there exists a Moore-Penrose grading g = $ gk such that p = $ gk is a Lie algebra of P. kGZ

k>O

One should be careful comparing graded and non-graded situations: if O c gk is a Moore-Penrose L-orbit then Ad(G)O c g is not necessarily a Moore-Penrose G-orbit. Let us give a criterion for an L-orbit to be MoorePenrose. Suppose that O = Ad(L)e c gk. Take any homogeneous sI2-triple (e, h, f ) . Then h defines a grading go = $ g t , such that ad(h);;. ,1 = n Id. nEZ

Theorem 8.49 ([T4]) O is a Moore-Penrose orbit if and only ifad(e)gt = 0 for any n > 0. In this case for any e1 E O there exists a unique homogeneous si2-triple (el, h', f l ) such that h' E ito. The proof is similar to the proof of Theorem 8.44.

Example 8.50 Suppose that G is a simple group of type Gz. We fix a root decomposition. There are two simple roots a1 and a 2 such that a1 is short and a 2 is long. There are 3 proper parabolic subgroups: the Borel subgroup B and two maximal parabolic subgroups PI and P2 such that a root vector of ai belongs to a Levi subgroup of Pi. Then the following is an easy application of Theorem 8.49. B is a Moore-Penrose parabolic subgroup (actually Borel subgroups in all simple groups are Moore-Penrose parabolic subgroups with respect to any grading). PI is not Moore-Penrose, but it is a MoorePenrose parabolic subgroup in degree 2 (with respect to the natural grading of height 2). P2 is a Moore-Penrose parabolic subgroup. Example 8.51 Suppose that G = SL,. We fix positive integers d l , . . . ,dk such that n = dl+. . .f dk. We consider the parabolic subgroup P(d1, . . . ,dk) C SLn that consists of all upper-triangular block matrices with sizes of blocks equal to dl,. . . ,dk. We take a standard grading. Then 01 is identified with the linear space of all tuples of linear maps { f l , . . . , fk),

0-1 is identified with the linear space of all tuples of linear maps {gl, . . . ,gk),

and Levi subgroup L(d1, . . . ,dk) is just the group (A1,. . . ,Ak) E GLdl x

. . . x GLdk

such that

det(A1) . . . . . det(Ak) = 1.

202


L(d1, . . . ,dk) 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 l , . . . ,mk-1 such that mi-1 mi L di (we set mo = mk = O), and consider the subvariety of all tuples {fl . . . ,fk-1) as above such that rk fi = mi and fi-1 o fi = 0 for any i. These tuples form a single L-orbit O called a variety of complexes. For any tuple {fl,. . . , f k P l ) E O consider the tuple {f?, . . . ,fz-,) E 8-1, where is a classical "matrix" Moore-Penrose inverse of fi. It is easy to see that this new tuple is again a complex and, 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.

+

fz

At first glance only few parabolic subgroups are Moore-Penrose. But this is scarcely true. For example, we have the following theorem:

Theorem 8.52 ([T4]) Any parabolic subgroup in SL, is Moore-Penrose. To explain our interest in Moore-Penrose parabolic subgroups let us recall Conjecture 8.30 from the previous section. Suppose once again that G is a simple connected simply-connected Lie group, P is its parabolic subgroup, and p c g are their Lie algebras. We take any irreducible G-module V. There exists a unique proper P-submodule Mv of V. We have the inclusion i : Mv -+ V, the projection n : V -+ V/MV and the map Rv : g -+ End(V) defining the representation. Therefore we have a linear map RV : g -+ Hom(Mv, VIMv), namely RV(X)= n o Rv(x) o i. Clearly p c Ker RV. Finally, we have a linear map Pv : g/p + Hom(Mv, VIMv). The Conjecture 8.30 states that there n exists an algebraic stratification g/p = ,U Xi such that for any V the function r=l rk Pv is constant along each Xi. The following theorem shows the connection of this problem with the Moore-Penrose inverse.

Theorem 8.53 ([T4]) Suppose that a grading g = @ gk is a MoorekEZ

Penrose grading in all positive degrees except at most one. Then the Conjecture 8.30 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 I , . . . ,x, E g let (xl,. . . ,x,),lg denote the minimal algebraic Lie subalgebra of g that contains X I , . . . ,x, (algebraic subalgebras are the Lie algebras of algebraic subgroups). By a theorem of Richardson [Rill (21,. . . , x , ) , ~ ~ is reductive if and only if an orbit of the r-tuple (xl, . . . ,x,) in gr is closed with respect to the diagonal action of G. Suppose now that hl, . . . , h, are semi-simple elements of g. Consider the closed subvariety 6 = (Ad(G)hl, . . . ,Ad(G) h,) c g'. For any closed G-orbit O c d let us denote by G(O) the conjugacy class of the reductive subalgebra (xl , . . . ,x,) ,lg for ( X ~ , - . . , X ,E) 0 .

8.6 Homogeneous Vector Bundles

203

Lemma 8.54 There are only finitely many conjugacy classes G ( 6 ) . 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 8 c g be the center of g, and g' c g be its commutant. Consider two canonical homomorphisms

We take any closed G-orbit 6 c d. Let (XI,.. . ,x,) E 6, 7 ~ i= n(xi) for i = 1,.. . , r. Then (yl,. . . ,Y , ) , ~ = ~ n((xl,. . . ,x,),lg) and is therefore reductive. Let us consider two cases. Suppose first that (yl, . . . ,yr)alg = 8'. Then g' is a commutor subgroup ~ therefore, ( X I , . . . ,x,),lg = (n' (hl), . . . ,n' (h,)) CBg'. In of (XI, . . . ,x , ) , ~ and, this case we get one conjugacy class. Suppose next that (yl, . . . ,yr),lg # 8'. Then (yl, . . . , y,),lg is contained in some maximal reductive Lie subalgebra of 8'. 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 fj' be one of them, fj = 3 @fj' c g. Let H be a corresponding reductive subgroup of G. It is sufficient to prove that, for any closed G-orbit 6 of d that meets fjr, there are only finitely many possibilities for G(6). It easily follows from Richardson's Lemma [Ri] that for any i the intersection Ad(G)hi n is a union of finitely many closed H-orbits, say A d ( ~ ) h : ,. . . ,Ad(H)h:i. It remains to prove that if, for some rtuple (XI,. . . ,x,) E Ad(~)h:l x . . . A ~ ( H:h) , the corresponding subalgebra (XI,. . . ,x,),~, is reductive then there are only finitely many possibilities for its conjugacy class. But this is precisely the claim of the lemma for the group H, which is true by the induction hypothesis. 0 Suppose that t is a compact real form of 0.

Lemma 8.55 If an r-tuple (XI,.. . ,x,) belongs to (it),, then its G-orbit is closed in gT.

Proof. Indeed, let B be a non-degenerate ad-invariant scalar product on g, which is negative-definite on t. Let H(x) = -B(?E, x) be a positive-definite P-invariant Hermitian quadratic form on g, where the complex conjugation is taken with respect to P. Let H r be a corresponding Hermitian quadratic form on gT. More precisely, HT(xl,.. . ,x,) = H(x1) . . . H(x,). By a KempfNess criterion [PV],in order to prove that the G-orbit of (XI,.. . ,x,) is closed, it is sufficient to prove that the real function Hr(.) has a critical point on this orbit. Let us show that (XI,.. . ,x,) is this critical point. Indeed, for any g E g

+ +

Now let G be a simple simply-connected Lie group, and let g be its Lie algebra with a Z-grading g = CB gk. Let r be a maximal integer such that ~ E Z

204


# 0. We denote the non-positive part of the grading

$ gk by p. Let P c G k50 be a parabolic subgroup with the Lie algebra p. We shall identify g/p with $ gk. Let L c G be a connected reductive subgroup with Lie algebra go. Let k>O V be an irreducible G-module. If we choose a Cartan subalgebra t C go then the grading of g originates from some Z-grading on tV.Therefore there exists a Z-grading V = $ Vk such that g i x c V,+j. Let R be a maximal integer

g,

~ E Z

such that VR # 0. It is easy to see that

Mv

= @I Vj (notice that VR is an

k

F

3

HnX.

L

We denote by O2 (a, b) the line bundle p*Ox (a) 63 0% (-bE). For example, Ox(F) = 02(1,2). By 7.11, we have 1 = (-n - d - 2)/2. Since x is the blowing up of X along L, K2 = O x j 1 , - n + d + 1). Let f : X 4 PN-l-d be the projection with center L. Let Y = f ( ~ ) . The hyperplane section H n X will correspond to a hyperplane section D of Y. Then f-l(D) = E F and f*Oy(l) = 02(l,1). Let y E Y \ D and Z = f-l(y). Suppose that d i m 2 2 1. Since Z n ( E U F) = 0, p maps Z isomorphically to a variety in X . Therefore 02(1, O)lz is non-trivial. But Ox(1, l)lz = f*Oy(l)lz is trivial. So 02(0, 1) is non-trivial. Hence Z n E # 0, which is a contradiction. Therefore, all positive-dimensional fibers of f belong to E n F , and in particular dimY = dimX. Now,

+

9.1 Smooth Self-dual Varieties

211

It follows from the Grauert-Rimenschneider vanishing theorem [GR] that

~ ~ ( O * ( a , l ) ) = ifo i > O

and a > ( n - 3 d - 2 ) / 2 .

(9.1)

~ ~ ( O % ( a ,= 2 0) ) if i > 0 and a 2 (n - 3d)/2.

(9.2)

Similarly,

Consider the exact sequence

Since H 1 ( 0 2 ( 0 ,2)) = 0, we have HO(OE(O,1 ) ) 2 HO(N:,,) = 0. Hence HO(N,V,,(a)) = 0 for a 5 0 and we have (i). Recall that NL,, = N x , X ( l ) .SO (ii) follows from (i) and Serre duality. Consider the exact sequence

By (9.1) and (9.2), we conclude that Hi(OE(a,1)) 21 Hi(N:,,(a)) = 0 for a > (n- 3d)/2 and we have (iii). (iv) follows from (iii) and Serre duality. Theorem 9.11 ([El]) Suppose that X is a smooth n-dimensional projective variety in PN with d = def X > 0. Let L = pd be a generic contact locus. Assume that K x = Ox(b) for some b G Z. Then def X

< ( n - 2)/2.

This inequality is strict unless dimX = 6,

def X = 2,

NX,, = H 1 (~:,,(-1)) @ O i ( 1 )

Proof. Consider the Beilinson spectral sequence with

which by Theorem 7.15 converges to NX,X. If d 2 ( n - 1)/2,then n - 3d - 2 5 d - n. It follows from Theorem 9.10 that Hq(Nx,x(p))= 0 for -d p 0. It follows that N;,X = 0. This is nonsense. If def X = (n-2)/2 then dimX = 4m+2 and def X = 2m by Theorem 7.4. In this case HQ(Nx,,(p)) = 0 for -2m 5 p 5 0 unless p = -m. This implies that Nx,, = Hm(N,V,,(-m)) 8 O p ( m ) . So

<
N/2+ 1. Then -N - 1 K x = ox(T ) by Example 7.11. We conclude that def X = N - 1 - n 5

n-2 2 Therefore, n = 2N/3. Now def X = N - 1 - n = - 1. Therefore n is even. n mod 2 by Theorem 7.4, we conclude that n = 4m 2 and Since def X def X = 2m. It follows that m 5 2 by Theorem 9.11. This leaves two cases: X is either a 6-dimensional variety in Pg or a 10-dimensionalvariety in PI5. 0

-by Theorem 9.11. Hence n 2 2N/3. By our assumption n 5 2N/3.

-

in

+

9.1.4 Finiteness Theorem Theorem 9.12 ([Munl]) Let p be a fixed positive integer. Then there exists No such that for any N > No the following holds. If X c PN is a smooth non-degenerate nonlinear n-fold and dim X * = n (p- 1) then either n > $ N (in which case X should be a complete intersection if Hartshorne's conjecture is true) or def X = k 2 n / 2 (in which case by Theorem 7.24 X is a projective bundle X 21 Py(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).

+

+ +

Proof. Let k < 1212 and n 5 $ N . Let L denote a contact locus of X with a generic tangent hyperplane. Then n > 2(N - p)/3 and for N 2 4p + 3 we have n > 1 N/2. Then PicX = Z is generated by O x ( l ) by Barth's Theorem [Ba]. If N > 9p - 6 then 4n 5 ;N < 3N - 3p 2, and therefore n-3k-2=n-3(N-p-n)-2 0 then this intersection is empty and we set d = 0. L can be considered as a pencil (one-dimensionallinear system) of divisors on X, and therefore it defines a rational map F : X --+ P1.More precisely, if L is spanned by hyperplanes H1 and H2 with equations f i = 0 and f2 = 0 then F has a form x ++(fi(x) : f2(x)) for any x E X. Thus F is not defined along the subvariety X n H1 n H2 of codimension 2. If we blow it up we get the variety x and the regular morphism fi : x -, P1,Let D C x be the exceptional divisor. Now let us calculate the topological Euler characteristic x ( ~ in ) two ways. First,

lx

220

10 Singularities of Dual Varieties

Here we use the fact that D is a P1-bundle over X n HI n Hz, and therefore x(D) = 2x(X n HI n Hz). Notice that the blowing up is an isomorphism on the complement of the exceptional divisor, and hence x(X \ D) = x ( X \ X n HI n HZ). On the other hand, we may use F to calculate x(x). For x E P1, we have x(F-~(x)) = x(X n Hz), where Hz is a hyperplane corresponding to x. Let X I , . . . ,xd E P1 be points that correspond to the intersection of L with X*. Then, for any x E P1 \ {xl,. . . ,xd), X n Hz is a smooth divisor. For any x E ($1,. . . ,xd), X n Hz has a simple quadratic singularity (see the proof of Theorem 7.1). It is clear then that x ( X n Hz) = x(X nHy) = x(X n H), where x, y E P1 \ { X I , . . . , xd) and H is a generic hyperplane. For x E { X I , . . . ,xd), near the simple quadratic singularity of X n Hz, the family of divisors X n Hy, y -+ x, looks like the family of smooth (n - 1)-dimensional quadrics Q , with = ET: near the unique singular point (0 : . . . : 0 : 1) equations T: + . . . + T:-l of the quadric Qo with the equation Tf . . . T:-l = 0. Therefore

+ +

Combining the two formulas for x(x), we get

+

The Euler characteristic of a k-dimensional smooth quadric is equal to k 2 for k even and to k 1 for k odd. The Euler characteristic 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 a (k - 1)-dimensional smooth quadric. Finally, we obtain the well-known class formula :

+

+

+

Theorem 10.1

Example 10.2 ([Kll, Lam]) The class formula is the main ingredient in the proof of the following formula of Landman (see [Lan, K131):

+

2 ( b n - l ( ~n H) - bn-1(x)) deg AX = (bn(x) - bn-z(x)) +(bn-z(x n H n H') - bn-2(X)), where bi 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

10.1 Class Formula

221

Example 10.3 Suppose that X c PN is a smooth projective curve of genus g and degree d. Let H c X be a hyperplane section. Then X * is a hypersurface by (1.19), and its degree is given by

where Kx denotes the canonical divisor. Example 10.4 ([Kll]) Let X c P2 be a smooth plane curve of degree d. Consider the Veronese embedding P2 c PN, where N 1 = (T;2), and the corresponding embedding X C PN. By the BBzout Theorem, X has degree d' = rd in this embedding. The dual variety X * c (PN)" parametrizes plane curves of degree r tangent to X at some point.

+

For example, the variety of conics tangent to a given conic is a hypersurface in P5 of degree 6. Example 10.5 Let X c PN be a smooth projective surface, and let H c X be a smooth hyperplane section. Then X * is a hypersurface by (7.5), and its degree is given by

deg A x = x(X) - 2x(H)

+ H . H = deg c2(T;) + 2Kx .H + 3 deg X.

Similar formulas exist if def X

> 0:

Theorem 10.6 ([Pa2]) Let X C PN be a smooth n-dimensional projective variety and let def X = k. Then

+ l)x(X) - (k + 2)x(X n H ) + x(X n Wk+2)], where H is a generic hyperplane and Wk+2 is a generic (k + 2)-codimensional deg X* = ( - I ) ~ +[(k ~

subspace. The topological Euler characteristic of a smooth variety X can be defined as x(X) = Jx c(Tx). If X is singular then one can substitute c(Tx) by the Chern-Mather class CM(X)defined as follows. Let x be the Nash blow-up of X , i.e. the closure in X x Gr(n, PN) of the graph of the Gauss map. Then x is equipped with the n-dimensional vector bundle S (the pullback of the tautological vector bundle of the Grassmannian) and CM (X) is defined as p,c(S) , where p : x -+ X is the projection. It turns out that the formula in Theorem 10.6 remains valid if we substitute the topological Euler characteristic by Jx CM(X).The codimension p part of the Chern-Mather class is equal to [Pi] P

ZM(X) =

C(-1 ) ~ - (n+13p+i j

j=O

222


where H is the hyperplane divisor and 17n-i c X is the codimension (n - i) polar variety, which is the closure of the set of smooth points of X at which the embedded tangent space Fz,xmeets a general linear space of codimension i + 2 not transversally. Ranks di (see Example 6.5) are equal to

In particular, ddef

= deg X*

Theorem 10.7 ([Erl]) For i = 0,. . . ,n we have

where xkis the intersection of X with generic k-codimensional linear subspace, and x is the topological Euler characteristic if X is smooth and the degree of the Cherr-Mather class if X is arbitrary.

10.2 Singularities of X * 10.2.1 Milnor Numbers Let f be a holomorphic function defined in some neighbourhood of 0 in C n . Suppose that 0 is an isolated singular point of the zero set Z(f) of f . Then the vector field (af /aa, . . . ,af 182,) has an isolated singularity at 0. The index of this vector field at 0 is called the Milnor number of Z(f) (or of f ) at 0 (see [Mi]). More generally, if M is a smooth algebraic variety and Z c M is a divisor with an isolated singular point p E Z then the Milnor number p(Z,p) is defined as a Milnor number of a local equation of Z at p. It is easy to calculate that if Z(f) has a simple quadratic singularity at 0 then the Milnor number is equal to 1. It follows that the Milnor number of f at 0 is equal to the number of critical points of a small perturbation of f near 0. Theorem 10.8 ([Dill) Let X C IPN be a smooth algebraic variety such that X* is a hypersurface. Let H E X* and suppose that (X n H),in, is finite. Then the multiplicity of X* at H is given by

Proof. Indeed, let U be a small neighbourhood of H in the complex topology, let Ho E U be a generic hyperplane, and let Ho+XF be a pencil of hyperplanes, where F is generic. If Ho is sufficiently close to H then this pencil intersects X*nU in m u l t ~ X* points. It follows that the function (more accurately, the

10.2 Singularities of X*

223

section of Ox(l)) H o / F has multH X * critical points near singular points of X n H . Indeed, critical points of Ho/F and singular points of divisors of the 0 form Ho + XF are, obviously, identical. Next we consider non-isolated singularities. Let M be a smooth projective variety, dim M = n. Let s E HO(M,L)\ (0) be a global section of a line bundle L over M , and let Z denote the zero set of s. We fix a Hermitian metric on L and consider the holomorphic component d of the corresponding metric connection. Let Y be a connected component of ZSing and let U be a small neighbourhood of Y with a smooth boundary I' in the complex topology. We may assume that ds does not vanish anywhere on r. Then the intersection index of ds with the zero section of the cotangent bundle of U tensored by L is called the Milnor number p(Z, Y). If Y is a point p then we get nothing else but the previous definition of the Milnor number. Similarly, we can define p(Z) as the intersection index of d s and the zero section of T g 8 C over a small neighbourhood of Z . Then p(Z) = C p(Z, Y,),where Yl,. . . ,Y, are all connected components of Zsing.

Theorem 10.9 ([Pal])

where x(M, L) is the expected Euler characteristic of the smooth zero set, i.e. of the zero set of a section transversal to the zero section. W e have

Theorem 10.8 has the following generalization.

Theorem 10.10 ([N]) Let X c PN be a smooth algebraic variety such that X * is a hypersurface. Let H E X * . Then

where Hg is a generic hyperplane.

If X n H has isolated singularities, i.e. (X r 7 H)si,g is 0-dimensional, then X n H n H, is smooth and we recover Theorem 10.8. If def X > 0 then the following analogue of this theorem is still true.

Theorem 10.11 ([Pa2]) Let X H E X * . Then

c PN

be a smooth algebraic variety, and let

where W g is a generic (dim X*)-dimensional subspace.

224


10.2.2 Milnor Class Suppose that M is a smooth n-dimensional algebraic variety, C is a line bundle on M , and X is the zero-scheme of a section of C. The singular scheme Sing X of X is the scheme supported on the singular locus of X and defined locally by the Jacobian ideal dF dF where X I , . X.

. .,x,

are local parameters for M and F is the section of L defining

Definition 10.12 Let Y be the singular scheme of a section of a line bundle C on a smooth variety M. The class

in the Chow group A.Y of Y is called the Milnor class of Y with respect to C. Here c denotes the total Chern class and s is the Segre class (in the sense of [Full). The following theorem summarizes the main properties of the Milnor class. Proofs and further details can be found in [All. Theorem 10.13 ([All) (i) pL(Y) only depends on Y and CIY: the Milnor class is independent of the choice of a smooth ambient variety M i n which Y is realized as the singular scheme of a section of a line bundle restricting t o CIy. (ii) Suppose that C is generated by global sections and let Xg be a generic section of C. Then

(iii) Suppose that Y is non-singular. Then

p ~ ( y= ) c(TVYCZJL) n [Y]. Example 10.14 If Y is supported on a point P, then pL(Y) = m[P], where m is the Milnor number of X at P. Indeed, in this case pL(Y) = s(Y, M), so m is the coefficient of [PI in s(Y, M), which agrees with the definition of the Milnor number (see [Full). Example 10.15 In general, let IY I be the support variety of Y, and assume that Y is proper. We claim that p(IY 1) defined before Theorem 10.9 is equal to the degree of the 0-dimensional component of pL(Y). Namely, the differential of the local equations of X in M determines a section of ( T V M8 C)(x.This can be extended to a holomorphic section s x of T V M8 L over the whole of M. Then p(IY1) is the contribution of (YI to the intersection of sx with


225

the zero section of T V M@ L. In the neighborhood of IYI we have the fiber diagram Y M

1

lax

M T*M@L that is, Y is the scheme-theoretic intersection of the two sections, so the contribution of IYI to the intersection number of the sections is equal (see [Full) to the degree of the 0-dimensional component of C(T~M 18 L) n s(Y, M),

which is the claim. In fact, in view of Theorem 10.13, the numerical information carried by the Milnor class of Y (when Y is proper) is equivalent to p-numbers of (YI and its generic sections IY n XgI, IX n Xg, n Xg21, etc. Now let us apply this machinery in our usual situation. Suppose that X c PN is a smooth projective variety, L = Ox(l), and X * c (PN)" is the dual variety parametrizing hyperplanes H such that the singular scheme (or the contact scheme in this case) Sing X n H is not empty. Then we have the Milnor class pL(Sing X nH) E A. (SingX nH). The problem that we address here is how to calculate the multiplicity of X * at H in terms of the Milnor class.

Theorem 10.16 ([Al, AC]) Let X n H be any singular hyperplane section of X . Then (i) The defect of the dual variety X * is the smallest integer r 2 0 such that cl (L)'c(L) n pr (Sing X n H)

# 0;

and for r = def X this number is equal to the multiplicity mHX*. (ii) If def X = r then mHX* is equal to

/.

(- 1)'pr (Sing X n H)

+ c1(L)'+'

n pc (Sing X n H).

Proof. (i) We argue by induction on r. If r = 0 then the integral is equal to

=

~((T'x b (7) g L) n s(Sing X n H, X).

Thinking of H as a point of X*, we have s(H, X*) = (mHX*)H, and thus mHX* = deg s(H, X*).

226


The dual variety is the projection from the conormal variety Ix,X* = ~F(Ix). Since r = 0, the degree of this projection is equal to 1. The fiber Y of T over H is identified with the contact locus Sing H n X (as a scheme). Therefore we have s(H, X*) = n,s(Y, Ix). Taking degrees, we get mHX* = deg s(Y, Ix). Now using the second projection Ix is quite easy to see that

-t

X, which is the projective bundle, it

s(Y, I*) = c(J(L)) n s(Sing X n H, X), where J(C) is the jet bundle of C. It remains to notice that c(J(L)) = c((TVX$0)8 L) by the exact sequence (5.1). Now let r > 0. By a similar argument to that above, we have

By Theorem 5.3, if X * is not a hypersurface then the dual variety XI* of a generic hyperplane section X' = X n Hg is the cone over X* with Hg as a vertex. The multiplicity of X* at H then evidently equals the multiplicity of XI* at H ; but def XI = def X - 1, and so the statement follows by induction because

=

/

/

CI (L)'c(L)

n pc(Sing X n H )

cl ( ~ ) ~ - l c ( Ln)(cl (C) n sing X =

] cl (L)'-'c(L)

n H))

n p r (Sing X' n H),

where we use Theorem 10.13 in the last equality. (ii) By (i) we see that def X 2 r if and only if the components of dimension i, 0 5 i < r, of c(L)npL(Sing X n H) vanish for all hyperplane sections X n H. That is, if def X 2 r, then for all hyperplane sections X n H

with Aj a class in dimension j, j = r, . . . ,dim Sing X Therefore

(with A,

# O), and

nH

(depending on H).

10.2 Singularities of X *

227

while

/

dim Sing XnH

CI (C)'+lP,c(sing

x nH ) =

( - I ) ~ - ~ + (~ ~C )~ni A,. i=r+l

Hence

which coincides with the formula in (i).

0

Remark 10.17 An easy Chern class computation shows that ranks (see Section 6.1) could be expressed via Milnor classes as follows:

Moreover, Theorem 6.2 can be viewed as a particular case of Theorem 10.16 applied to the zero-section of HO(X,C). Theorem 7.4 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 dimX. 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 [All.

Theorem 10.18 ([All) Let X c PN be a smooth projective variety. Let H E X* be such that the contact locus Y = Sing X n H is pure-dimensional and non-singular i n a neighborhood of some complete curve C c Y of odd degree. Then dim Y = dim X mod 2. For example, if Y is a generic contact locus then Y = pdefX . Therefore, if Y is not a point then it contains a line C and we get Theorem 7.4. The following result is an immediate corollary of Theorem 10.16:

Theorem 10.19 ([All) Let X c PN be a smooth projective variety. Suppose that the contact scheme Sing X n H of a hyperplane H with X is a projective space Pr. Then def X is equal to r and X* is smooth at H . Recall that by a contact scheme we mean the contact locus endowed with the scheme structure given by the Jacobian ideal. The surprising feature of this result is that we are not assuming a hyperplane to be generic. The dimension of the contact scheme of a hyperplane is not necessarily equal to the dimension of the contact locus of a generic hyperplane. The claim, however, is that it is

228


so if the contact scheme is a projective subspace as a scheme. Moreover, it can be shown (also using Theorem 10.16) that if the contact scheme of a variety with a given hyperplane is smooth then the defect of the variety is equal to the defect of the contact scheme. This 'explains' the previous theorem, because the defect of a projective subspace is equal to its dimension. If def X = 0, Theorem 10.19 can be reformulated a little more explicitly. Let X c Pn be a smooth variety such that X* is a hypersurface. Let I X c Pn x (Pn)" be a conormal variety. We denote by n the projection Ix -+ X*. Let U C X * be the set of points where n is unramified. In other words, H E U iff Sing X n H is a single point p as a scheme (the Hessian of a function defining X n H is non-zero at p). Clearly, U is non-empty and open in X*.

Theorem 10.20 ([Ka]) Suppose that X c IPN is smooth and X * is a hypersurface. Then U consists of smooth points of X* and is the biggest open set i n X * for which the projection n : n-l(U) -+ U is an isomorphism. In fact, one may expect that U = X,,. has codimension 1.

For example, this is so if X * \ U

Proposition 10.21 Suppose that X c PN is smooth and X* is a hypersurface. If Z = X* \ U is a divisor then Z = (X*),ing. Proof. By Theorem 10.20 X*,ing c 2. We need to prove an opposite inclusion. Let Y c X * be the variety of all points x E X* such that dimn-l(x) > 0. Then Y n U = 0, and therefore codimp Y 2 2. If Zo c Z is an irreducible component then for generic point H E Zo the contact locus Sing H n X consists of finitely many points. Therefore, H E (X*),ing, e.g. by Theorem 10.8. 10.2.3 Dual Variety of a Surface Let X be a smooth projective algebraic surface with a very ample line bundle L. By 7.5, X* c JCJis a hypersurface. We want to find m D X * for each D E IL(.Suppose that 7-

where any curve Di is reduced and irreducible. Let Dred =

Di.

Theorem 10.22 ([AC]) With the above notation, we have

where K x is the canonical divisor o f X and p is the ordinary Milnor number.


229

Proof. Take a general H E I,C( and denote by L C (L(the pencil containing D and H. Then (10.1) mDX* = degX* - s, where s is the number of singular members of L different from D (each of these 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 [GHl]. H . Di) the points of interFor each i (1 5 i r ) denote by pij (1 j section of H and Di. Let x denote the surface obtained from X by blowing up each point pij ni times (in the direction of H). The induced pencil on x is base-point-free and gives a map f : x -, IP1. If E:, k = 1,. . . ,nil are the exceptional divisors at pij then the fiber of f at the point 0 (corresponding to D) is (ni-k)~&. ~*(o)=D+C

3, X*,ing has three irreducible components X*cusp,X*node(!d),and X*,,de(3). For 2 x 2 x 2 x 2 matrices, X*,ing has eight irreducible components X*cuspl X*node(Q))tand X*node(i,j) for 1 i i < j 5 4. For m x m x 2 x 2 matrices with m > 2, X*,ing has three irreducible ~ 0 m p 0 n e nX*cusp, t~ X*node(Q)),and X*node(3,4). In all cases all irreducible components O ~ X * , have ~ , ~codimension 1 unless written otherwise.

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Index

.

.

Hes 57 L a m 7 Ox 13

&(x)

80 Tan(Y,X) 81 ad 19 def X 9 X* 2

246

Index

adjoint variety 188 admissible collection Albert algebra 44 algebraic group 17 torus 22 Aluffi VIII, 219 ample 15 annihilator 172 aura 196

101

BBzout formula 107 base-point free 15 based exact complex 102 Beilinson spectral sequence 124 Beltrametti VII bitangent 4 Bore1 subgroup 27 Borel-Weyl-Bott Theorem 155 boundary format 167 capacity of a network 52 Cartan Lemma 66 subalgebra 22 Cartier divisor 13 Cauchy formula 158 caustic 3 curve 3 point 3 Cayley VII, 99 determinant 99 method 109 numbers 44

trick 88,89,96 characteristic of a nilpotent 199 Chern class 110 polynomial 111 Chow form 88 group 110 class formula 220 of a plane curve 4 classical discriminant 28,103,112,153 resultant 96 Clebsch 4 Clifford algebra 30 Coble quartic 213 cohomology of Koszul complexes 98 complete linear system 15 Concini 109 cone of effective 1-cycles 137 conical action 32 orbit 32 connected component 18 conormal bundle 7 variety 7 contact locus 10,119 scheme 227 Contraction Theorem 138 correspondence of branches 4 cotangent bundle 7 Cukierman VIII, 219 cusp 4 cuspidal point see cusp cut of a network 52 Darboux frame 66 De Concini VII defect of a projective variety 9 degeneracy locus 134 degree of a cycle 110 Del Pezzo fibration 145 surface 182 derivations 21 desingularization of a dual variety

10

Index determinant of a complex 101 of a supervector space 99 of a vector space 99 digraph 52 Dimca VIII, 219 discriminant 9 complex 102 of a binary form 9 of a linear operator 186 of an element in a Lie algebra 187 spectral sequence 105 distinguished nilpotent orbit 216 dominant weight 26 doubling 131 dual of the curve 11 of the smooth surface 121 plane curve 2 quiver 50 representation 19 variety 2 variety of a smooth complete intersection 112 vector space 1 effective 1-cycle 136 divisor 13 Ein VI, 89,119,207 Theorem 119,210 embedded projective tangent space equivaxiant vector bundle 156 Ernstrom VIII, 219 Euler characteristic 115 Euler-Meusnier Theorem 67 extremal rational curve 138 ray 137 Fania VII, 168 Fano variety 140 Finiteness Theorem 212 first jet 97 Fischer 99 flag variety 27 flex see inflection point flow in a network 52

2

Franz 99 Fulton VI, 73 fundamental form 64 weights 147 Gauss image 73 map 73 Gauss-Bonnet formula 219 Gelfand V, 89 generic rank 109 Grobner basis 162 Grassmannian 28 fibration 87 Griffiths 57 Grothendieck ring 109 group of norm similarities 44,45 half-spinor module 31 Hansen VI, 73 Harris 57 Hartshorne 94,209 conjecture 83,209 height function 38 of a nilpotent orbit 199 Hessian matrix 57 higher fundamental forms 68 highest root 26 weight 26 Hilbert polynomial 115 scheme 141 syzygy theorem 109 Holme VI, 89,109 homogeneous sI2-triple 197 real hypersurface 209 space 26 horizontal 1-form 66 hyperdeterminants 63,166 index of a Fano variety 140 inflection point 4 intersection product 136 Ionescu 168 isotropic

247

248

Index

Grassmannian subspace 28

29

Jacobian ideal 224 jet bundle 97 join 74 Jordan algebra 44 decomposition 20, 172 Kapranov V, 89 Katsylo VIII Katz VII, 57, 109 dimension formula 57 Katz-Kleiman-Holme formula 111 Keel VIII Kempf-Ness criterion 203 Killing form 21 Kleiman VII, 109 criterion 137 Knop VII, 119 Kobayashi 145 Kodaira dimension 123 Koszul complex 97 Kronecker-Weierstrass theory 131 Kummer surface 213 Kuznetsov VIII Lagrangian subvariety 7 Landman VII Landman formula 220 Landsberg 57 Lascoux VII, 109 Legendre transformation 3 length function 25 of extremal rays 139 Leung VIII Levi decomposition 156 subgroup 156 linear algebraic group 18 system of divisors 15 linearly equivalent divisors 14 normal variety 13 Livorni 168 local

equations of a divisor long root 25 lowest weight 26

13

Matsumura-Monsky Theorem Maurer-Cartan equations 65 forms 65 maximal parabolic subgroup 27 torus 22 Menzel VII, 119 Milnor class 224 number 222,224 minimal flag variety 27 moment map 33 monad 126 Monotonicity Theorem 124 MoorePenrose inverse 196 orbit 201 parabolic subgroup 201 Mori Cone Theorem 138 moving frames 65 Mukai fibration 145 multiple tangent 4 multisecant variety 75 multisegment 50 duality 50 Munoz VI, 89,207 nef divisors 137 value 138 value morphism 139 Nemethi VIII, 219 network 52 nilpotent cone 26,216 element 26 node 4 non-degenerate variety 12 norm similarities 44 normal bundle 7 normalizer 18 numerically equivalent 136

169

Index trivial

136

Ochiai 145 octonions 44 Okonek 168 order of a projective variety 75 ordinary double point see node osculating sequence 67 space 67,68 parabolic subgroup 27 parity theorem 121 path in a network 52 pencils of symmetric matrices 131 permanent 160 Picard number 136 Pliicker embedding 28,71 formulas 4 polarized variety 15, 136 Popov VIII, 207 Product Theorem 60 projection from a subspace 11 projective extension 93 second fundamental form 64 space 1 projectively dual subspace 1 dual variety 2 Pyasetskii Theorem 32 quadratic nilpotent 171 quadric fibration 145 quasi-annihilator 172 quasiderivation 173 quotient projective space 11 radical 20 Ran VI, 73 rank of a projective variety 113 of a reductive group 22 rational normal curve 9 Rationality Theorem 138

249

reductive group 21 Reflexivity Theorem 2 , 6 regular algebra 172 semisimple operator 186 Reidermeister torsion 99 relative tangential variety 81 representation of a quiver 50 resultant complexes 105 variety 96 Richardson 202 Riemann-Roch-Hirzebruch theorem 115 root lattice 147 ruled variety 10 Saltman VIII Schlafli formula 5 scroll 129, 145 secant 75 variety 75 Segre embedding 39,60,68 Theorem 59 self-dual nilpotent orbit 215 variety 207 semi-ample divisor 136 semisimple element 20 group 21 operator 20 Severi variety 45,72,83 short grading 41 root 25 simple roots 23 singular scheme 224 subalgebra 171 weight 156 smooth extendability 93 Snow VII, 119 Sommese VII spectral sequence of the double complex 104

250

Index

spinor group 30 variety 29,71 stabilizer 18 stable point 169 stably twisted discriminant complex 102 standard invariant form 21 supervector space 99 Swinnerton-Dyer 168 Sylvester formula 103,106 normal form 186 pentahedron 182 symplectic structure 7 variety 7 symplectic structure on the cotangent bundle 8 tangent 75 bundle 7 hyperplane 2 star 81 tangential variety 80 tautological line bundle 95 tautological vector bundle Terracini lemma 74 Theorem

Printing: Mercedes-Druck, Berlin Binding: Stein + Lehmann, Berlin

28

on tangencies 81 Theorem on Severi varieties uniform vector bundle unipotent element 20 group 20 operator 20 radical 20

83

121

variety of complexes 202 of the highest weight vectors 27 with a small dual variety 145 Veronese embedding 28,69 very ample 15 vector bundle 96 Vinberg VIII weight of a multisegment 50 system 25 Weil 168 Weyl chamber 25,156 Weyman VI, 57,109, 219 Zak VI, 73 theorems 81 Zelevinsky V, 57,89, 219 involution 50

Projective Duality and Homogeneous Spaces

Projective Duality and Homogeneous Spaces

Projective and polar spaces

Homogeneous Spaces and Equivariant Embeddings

Homogeneous Spaces and Equivariant Embeddings

Homogeneous Finsler spaces

Algebraic Homogeneous Spaces and Invariant Theory

Harmonic Analysis on Homogeneous Spaces

Algebraic homogeneous spaces and invariant theory

Harmonic Analysis on Spaces of Homogeneous Type

Classical homogeneous spaces (2006)(en)(14s)

Projective geometry and projective metrics

Harmonic analysis on spaces of homogeneous type

Duality

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Duality

Vector bundles on complex projective spaces

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Mirror Geometry of Lie Algebras, Lie Groups, and Homogeneous Spaces

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Bundles of Topological Vector Spaces and Their Duality

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Mirror Geometry of Lie Algebras, Lie Groups and Homogeneous Spaces

Valuations of Skew Fields and Projective Hjelmslev Spaces

Foundations of Incidence Geometry: Projective and Polar Spaces

Residues and duality

Cones and Duality

Duality and Modern Economics

Interpolation Functors and Duality

Duality and Modern Economics

Duality and supersymmetric theories

Projective Duality and Homogeneous Spaces