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Evgueni Tevelev Projectively Dual Varieties arXiv:math/0112028v1 [math.AG] 3 Dec 2001 Preface During several centuries various reincarnations of projective duality have in- spired research in algebraic and differential geometry, classical mechanics, invariant theory, combinatorics, etc. On the other hand, projective duality is simply the systematic way of recovering the projective variety from the set of its tangent hyperplanes. In this survey we have tried to collect together different aspects of projective duality and points of view on it. We hope, that the exposition is quite informal and requires only a standard knowledge of algebraic geometry and algebraic (or Lie) groups theory. Some chapters are, however, more difficult and use the modern intersection theory and homology algebra. But even in these cases we have tried to give simple examples and avoid technical difficulties. An interesting feature of projective duality is given by the observation that most important examples carry the natural action of the Lie group. This is especially true for projective varieties that have extremal properties from the point of view of projective geometry. We have tried to stress this phenomenon in this survey and to discuss many variants of it. However, one aspect is completely omitted – we are not discussing the dual varieties of toric varieties and the corresponding theory of A-discriminants. This theory is presented in the beautiful book [GKZ2] and we feel no need to reproduce it. Parts of this survey were written during my visits to the Erwin Shroedinger Institute in Vienna and Mathematic Institute in Basel. I would like to thank my hosts for the warm hospitality. I have discussed the contents of this book with many people, including E. Vinberg, V. Popov, A. Kuznetsov, S. Keel, H. Kraft, D. Timashev, D. Saltman, P. Katsylo, and learned a lot from them. I am especially grateful to F. Zak for providing a lot of information on pro- jective duality and other aspects of projective geometry. Edinburgh, November 2001 Evgueni Tevelev II Preface Table of Contents 1. Dual Varieties ............................................ 1 1.1 Definitions and First Properties . 1 1.2 ReflexivityTheorem .................................. .. 4 1.2.A TheConormalVariety ............................ 4 1.2.B Applications of the Reflexivity Theorem . 6 1.3 DualPlaneCurves................................... ... 8 1.3.A Parametric Representation of the Dual Plane Curve .. 8 1.3.B The Legendre Transformation and Caustics .......... 9 1.3.C Correspondence of Branches. Pl¨ucker Formulas. .. .. 10 1.4 Projections and Linear Normality . 12 1.4.A Projections ...................................... 12 1.4.B LinearNormality................................. 14 1.5 Dual Varieties of Smooth Divisors . 17 2. Dual Varieties of Algebraic Group Orbits ................. 21 2.1 Polarized Flag Varieties . 21 2.1.A Definitions and Notations ......................... 21 2.1.B BasicExamples .................................. 23 2.2 The PyasetskiiPairing ................................ .. 30 2.2.A Actions With Finitely Many Orbits . 30 2.2.B The Multisegment Duality. 32 2.3 Parabolic Subgroups With Abelian Unipotent Radical . 34 3. The Cayley Method for Studying Discriminants .......... 41 3.1 Jet Bundles and Koszul Complexes ....................... 41 3.2 Cayley Determinants of Exact Complexes ................. 43 3.3 Discriminant Complexes . 46 4. Resultants and Schemes of Zeros ......................... 51 4.1 AmpleVector Bundles ................................. 51 4.2 Resultants........................................ ..... 52 4.3 Zeros of Generic Global Sections ........................ 54 4.4 Moore–Penrose Inverse and Applications ................. 61 IV Table of Contents 5. Fulton–Hansen Theorem and Applications ................ 71 5.1 Fulton–Hansen Connectedness Theorem ................. .. 71 5.2 Secant and Tangential Varieties ........................ .. 74 5.3 ZakTheorems ...................................... ... 76 6. Dual Varieties and Projective Differential Geometry ...... 81 6.1 TheKatzDimensionFormula............................ 81 6.2 Product Theorem and Applications ....................... 84 6.2.A ProductTheorem ................................ 84 6.2.B Hyperdeterminants ............................... 87 6.2.C Associated Hypersurfaces.......................... 88 6.3 EinTheorems....................................... ... 89 6.4 The Projective Second Fundamental Form ................ 96 6.4.A GaussMap...................................... 96 6.4.B MovingFrames .................................. 98 6.4.C Fundamental Forms of Projective Homogeneous Spaces 101 7. The Degree of a Dual Variety ............................. 105 7.1 Katz–Kleiman–Holme Formula . 105 7.2 Degrees of Discriminants of Polarized Flag Varieties . 108 7.2.A The Degree of the Dual Variety to G/B ............. 108 7.2.B Degrees of Hyperdeterminants ..................... 110 7.2.C One Calculation . 111 7.3 Degree of the Discriminant as a Function in .............. 116 7.3.A General Positivity Theorem . .L . 117 7.3.B Applications to Polarized Flag Varieties . 118 7.4 Gelfand–Kapranov Formula............................ .. 119 8. Milnor Classes and Multiplicities of Discriminants ........ 121 8.1 ClassFormula....................................... ... 121 8.2 MilnorClasses ....................................... .. 122 8.3 Applications to Multiplicities of Discriminants . 124 8.4 Multiplicities of the Dual Variety of a Surface . 126 8.5 Further Applications to Dual Varieties . 128 9. Mori Theory and Dual Varieties .......................... 133 9.1 Some Results From Mori Theory ......................... 133 9.2 Mori Theory and Dual Varieties......................... 138 9.2.A The Nef Value and the Defect...................... 138 9.2.B The Defect of Fibers of the Nef Value Morphism ..... 140 9.2.C Varieties With Small Dual Varieties. 142 9.3 Polarized Flag Varieties With Positive Defect . 144 9.3.A Nef Value of Polarized Flag Varieties . 144 9.3.B Classification . 147 Table of Contents V 10. Some Applications of the Duality ......................... 151 10.1 Discriminants and Automorphisms. 151 10.1.AMatsumura–Monsky Theorem...................... 151 10.1.B Quasiderivations of Commutative Algebras . 153 10.2 Discriminants of Anticommutative Algebras . 157 10.3 Self–dual Varieties................................. ..... 168 10.3.A Self–dual Polarized Flag Varieties . 168 10.3.B Around Hartshorne Conjecture..................... 170 10.3.C Self–dual Nilpotent Orbits . 172 10.4 Linear Systems of Quadrics of Constant Rank............. 172 References ................................................... 178 Index ................................................... ...... 187 1. Dual Varieties Preliminaries The projective duality gives a remarkably simple method to recover any pro- jective variety from the set of its tangent hyperplanes. In this chapter we recall this classical notion, give the proof of the Reflexivity Theorem and its consequences, provide a number of examples and motivations, and fix the notation to be used throughout the book. The exposition is fairly standard and classical. In the proof of the Reflexivity Theorem we follow [GKZ2] and deduce this result from the classical theorem of symplectic geometry saying that any conical Lagrangian subvariety of the cotangent bundle is equal to some conormal variety. 1.1 Definitions and First Properties For any finite-dimensional complex vector space V we denote by P(V ) its projectivization, that is, the set of 1-dimensional subspaces. For example, if V = Cn+1 is a standard complex vector space then Pn = P(Cn+1) is a standard complex projective space. A point of Pn is defined by (n + 1) homogeneous coordinates (x0 : ... : x1), xi C, which are not all equal to 0 and are considered up to a scalar multiple. ∈ If U V is a non-trivial linear subspace then P(U) is a subset of P(V ), subsets of⊂ this form are called projective subspaces. Projective subspaces of dimension 1, 2, or of codimension 1 are called lines, planes, and hyperplanes. For any vector space V we denote by V ∗ the dual vector space, the vector space of linear forms on V . Points of the dual projective space P(V )∗ = P(V ∗) correspond to hyperplanes in P(V ). Conversely, to any point p of P(V ), we can associate a hyperplane in P(V )∗, namely the set of all hyperplanes in P(V ) passing through p. Therefore, P(V )∗∗ is naturally identified with P(V ). Of course, this reflects nothing else but a usual canonical isomorphism V ∗∗ = V . To any vector subspace U V we associate its annihilator Ann(U) V ∗. Namely, Ann(U) = f V ∗ ⊂f(U)=0 . We have Ann(Ann(U)) = U⊂. This corresponds to the projective{ ∈ | duality between} projective subspaces in P(V ) and P(V )∗: for any projective subspace L P(V ) we denote by L∗ P(V )∗ its dual projective subspace, parametrizing⊂ all hyperplanes that contain⊂ L. 2 1. Dual Varieties Remarkably, the projective duality between projective subspaces in Pn and Pn∗ can be extended to the involutive correspondence between irreducible algebraic subvarieties in Pn and Pn∗. First, suppose that X Pn is a smooth irreducible algebraic subvariety. ⊂ n For any x X, we denote by TˆxX P an embedded projective tangent space. More∈ precisely, if X P(V ) is⊂ any projective variety then we define the cone Cone(X) V over∈ it as a conical variety formed by all lines l such that P(l) X. If ⊂x X is a smooth point then any non-zero point v of ∈ ∈ the corresponding line is a smooth point of Cone(X) and
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