Projectively dual varieties

195 Pages·2001·1.129 MB·English

by  Tevelev E.

Evgueni Tevelev
Projectively Dual Varieties
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
differentaspectsofprojectivedualityandpointsofviewonit.Wehope,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,moredifficultandusethemodernintersectiontheoryandhomology
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.
PartsofthissurveywerewrittenduringmyvisitstotheErwinShroedinger
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,andlearnedalotfromthem.
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 Reflexivity Theorem .................................... 4
1.2.A The Conormal Variety ............................ 4
1.2.B Applications of the Reflexivity Theorem............. 6
1.3 Dual Plane Curves...................................... 8
1.3.A Parametric Representation of the Dual Plane Curve .. 8
1.3.B The Legendre Transformation and Caustics.......... 9
1.3.C Correspondence of Branches. Plu¨cker Formulas. ...... 10
1.4 Projections and Linear Normality ........................ 12
1.4.A Projections ...................................... 12
1.4.B Linear Normality................................. 14
1.5 Dual Varieties of Smooth Divisors ........................ 17
2. Dual Varieties of Algebraic Group Orbits................. 21
2.1 Polarized Flag Varieties ................................. 21
2.1.A Definitions and Notations ......................... 21
2.1.B Basic Examples .................................. 23
2.2 The Pyasetskii Pairing .................................. 30
2.2.A Actions With Finitely Many Orbits................. 30
2.2.B The Multisegment Duality. ........................ 32
2.3 Parabolic Subgroups With Abelian Unipotent Radical....... 34
3. The Cayley Method for Studying Discriminants .......... 41
3.1 Jet Bundles and Koszul Complexes ....................... 41
3.2 Cayley Determinants of Exact Complexes ................. 43
3.3 Discriminant Complexes................................. 46
4. Resultants and Schemes of Zeros ......................... 51
4.1 Ample Vector Bundles .................................. 51
4.2 Resultants............................................. 52
4.3 Zeros of Generic Global Sections ......................... 54
4.4 Moore–Penrose Inverse and Applications .................. 61
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 Zak Theorems ......................................... 76
6. Dual Varieties and Projective Differential Geometry...... 81
6.1 The Katz Dimension Formula ............................ 81
6.2 Product Theorem and Applications ....................... 84
6.2.A Product Theorem ................................ 84
6.2.B Hyperdeterminants ............................... 87
6.2.C Associated Hypersurfaces.......................... 88
6.3 Ein Theorems.......................................... 89
6.4 The Projective Second Fundamental Form ................. 96
6.4.A Gauss Map ...................................... 96
6.4.B Moving Frames .................................. 98
6.4.C Fundamental Forms of Projective Homogeneous Spaces 101
7. The Degree of a Dual Variety............................. 105
7.1 Katz–Kleiman–Holme Formula ........................... 105
7.2 Degrees of Discriminants of Polarized Flag Varieties ........ 108
7.2.A The Degree of the Dual Variety to G/B ............. 108
7.2.B Degrees of Hyperdeterminants ..................... 110
7.2.C One Calculation.................................. 111
7.3 Degree of the Discriminant as a Function in L.............. 116
7.3.A General Positivity Theorem........................ 117
7.3.B Applications to Polarized Flag Varieties ............. 118
7.4 Gelfand–Kapranov Formula.............................. 119
8. Milnor Classes and Multiplicities of Discriminants ........ 121
8.1 Class Formula.......................................... 121
8.2 Milnor Classes ......................................... 122
8.3 Applications to Multiplicities of Discriminants ............. 124
8.4 Multiplicities of the Dual Variety of a Surface .............. 126
8.5 Further Applications to Dual Varieties .................... 128
9. Mori Theory and Dual Varieties .......................... 133
9.1 Some Results From Mori Theory ......................... 133
9.2 Mori Theory and Dual Varieties .......................... 138
9.2.A The Nef Value and the Defect...................... 138
9.2.B The Defect of Fibers of the Nef Value Morphism ..... 140
9.2.C Varieties With Small Dual Varieties................. 142
9.3 Polarized Flag Varieties With Positive Defect .............. 144
9.3.A Nef Value of Polarized Flag Varieties................ 144
9.3.B Classification .................................... 147
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.BQuasiderivations of Commutative Algebras .......... 153
10.2 Discriminants of Anticommutative Algebras................ 157
10.3 Self–dual Varieties...................................... 168
10.3.ASelf–dual Polarized Flag Varieties .................. 168
10.3.BAround Hartshorne Conjecture..................... 170
10.3.CSelf–dual Nilpotent Orbits......................... 172
10.4 Linear Systems of Quadrics of Constant Rank.............. 172
References.................................................... 178
Index......................................................... 187
1. Dual Varieties
Preliminaries
Theprojectivedualitygivesaremarkablysimplemethodtorecoveranypro-
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 (x :...:x ), x ∈C, which are not all equal to 0
0 1 i
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
dimension1, 2,orofcodimension1 arecalledlines,planes,and hyperplanes.
ForanyvectorspaceV wedenotebyV∗ thedualvectorspace,thevector
spaceoflinearformsonV.PointsofthedualprojectivespaceP(V)∗ =P(V∗)
correspondtohyperplanesinP(V).Conversely,toanypointpofP(V),wecan
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,thisreflectsnothingelsebutausualcanonicalisomorphismV∗∗ =V.
ToanyvectorsubspaceU ⊂V weassociateitsannihilatorAnn(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
andPn∗canbeextendedtotheinvolutivecorrespondencebetweenirreducible
algebraic subvarieties in Pn and Pn∗.
First, suppose that X ⊂ Pn is a smooth irreducible algebraic subvariety.
For any x ∈ X, we denote by Tˆ X ⊂ Pn an embedded projective tangent
x
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 Tˆ (X) is defined
x
as P(T Cone(X)), where T Cone(X) is a tangent space of Cone(X) at v
v v
consideredasalinearsubspaceofV (itdoesnotdependonachoiceofv).For
any hyperplane H ⊂ Pn, we say that H is tangent to X at x if H contains
Tˆ X. We define the dual variety X∗ ⊂ Pn∗ as the set of all hyperplanes
x
tangent to X.
In other words, a hyperplane H belongs to X∗ if and only if the inter-
section X ∩H (regarded as a scheme) is singular (is not a smooth algebraic
variety). In most parts of this book we shall be interested only in dual vari-
etiesofsmoothvarieties,sothisdescriptionofX∗ willbesufficient.However,
with this definition we can not expect the duality X∗∗ =X because X∗ can
besingular(andinmostinterestingcasesitisactuallysingular).Soweshould
define X∗ for a singular X as well. There are in fact two possibilities: first,
wecanimitatethepreviousdefinitionandconsiderembeddedtangentspaces
at all points, not necessarily smooth. But it turns out that the dual variety
defined in this fashion does not have good properties, for example it can be
reducible. The better way is to pick only the ‘main’ component of the dual
variety.
Definition 1.1 Let X ⊂ Pn be an irreducible projective variety. A hyper-
plane H ⊂ Pn is called tangent to X if it contains an embedded tangent
spaceTˆ X atsomesmoothpointx∈X.Theclosureofthesetofalltangent
x
hyperplanes is called the dual variety X∗ ⊂Pn∗.
We shall discuss the Reflexivity Theorem X∗∗ =X and its consequences
in the next section. First we shall establish some simple properties of dual
varieties and give further definitions.
Definition 1.2 Let X ⊂ Pn be an irreducible projective variety with the
smooth locus X . Consider the set I0 ⊂Pn×Pn∗ of pairs (x,H) such that
sm X
x∈X and H is the hyperplane tangent to X at x. The Zariski closure I
sm X
of I0 is called the conormal variety of X.
X
The projection pr : I0 →X makes I0 into a bundle over X whose
1 X sm X sm
fibers are projective subspaces of dimension n−dimX −1. Therefore, I0
X
and I are irreducible varieties of dimension n−1. By definition, X∗ is the
X
image of the projection pr : I → Pn∗. Therefore, we have the following
2 X
proposition:
1.1 Definitions and First Properties 3
Proposition 1.3 X∗ is an irreducible variety.
Moreover, since dimI =n−1, we can expect that in ‘typical’ cases X∗
X
is a hypersurface. Having this in mind, we give the following definition:
Definition 1.4 The number codim X∗−1 is called the defect of X, de-
Pn∗
noted by defX.
Typically, defX = 0. In this case X∗ is defined by an irreducible homo-
geneous polynomial ∆ .
X
Definition 1.5 ∆ is called the discriminant of X.
X
IfdefX >0then,forconvenience,weset∆ =1.Clearly,∆ isdefined
X X
only up to a scalar multiple. Roughly speaking, the study of dual varieties
and discriminants includes the following 3 steps:
– To define some nice natural class of projective varieties X.
– To find all exceptional cases when defX >0.
– In the remaining cases, to say something about ∆ or X∗.
X
Inthelaststeptheminimalprogramistodeterminethedegreeof∆ ,the
X
maximalprogramistodetermine∆ asapolynomial,oratleasttodescribe
X
itsmonomials.AnotherinterestingproblemistodescribesingularitiesofX∗.
Example 1.6 Themostfamiliarexampleofadiscriminant∆ is,ofcourse,
X
the discriminant of a binary form. In order to show that it actually coincides
with some ∆ we need first to give an equivalent definition of ∆ . Suppose
X X
that x ,...,x are some local coordinates on Cone(X) ⊂ V. Any f ∈ V∗, a
1 k
linearformonV,beingrestrictedtoCone(X)becomesanalgebraicfunction
in x ,...,x . Then ∆ is just an irreducible polynomial, which vanishes at
1 k X
f ∈ V∗ whenever the function f(x ,...,x ) has a multiple root, that is,
1 k
vanishes at some v ∈ Cone(X), v 6= 0, together with all first derivatives
∂f/∂x .
i
Consider now the d-dimensional projective space Pd = P(V) with homo-
geneous coordinates z ,...,z , and let X ⊂Pd be the Veronese curve
0 d
(xd :xd−1y :xd−2y2 :...:xyd−1 :yd), x,y ∈C, (x,y)6=(0,0)
(the image of the Veronese embedding P1 ⊂ Pd). Any linear form f(z) =
(cid:80)
a z is uniquely determined by its restriction to the cone Cone(X), which
i i (cid:80)
is a binary form f(x,y)= a xd−iyi. Therefore, f ∈Cone(X∗) if and only
i
if f(x,y) vanishes at some point (x ,y ) 6= (0,0) (so (x : y ) is a root of
0 0 0 0
f(x,y)) with its first derivatives (so (x :y ) is a multiple root of f(x,y)). It
0 0
follows that ∆ is the classical discriminant of a binary form.
X
4 1. Dual Varieties
1.2 Reflexivity Theorem
In this section we shall prove the Reflexivity Theorem. We follow the expo-
sition in [GKZ2], which shows that there exists a deep connection between
projective duality and symplectic geometry. Other proofs, including the in-
vestigation of a prime characteristic case, could be found, e.g., in [Se], [M],
[Wa].
Theorem 1.7
(a) For any irreducible projective variety X ⊂Pn, we have X∗∗ =X.
(b) More precisely, If z is a smooth point of X and H is a smooth point of
X∗,thenH istangenttoX atz ifandonlyifz,regardedasahyperplane
in Pn∗, is tangent to X∗ at H.
The proof will be given in the next section.
1.2.A The Conormal Variety
We shall need some standard definitions.
Definition 1.8 If X is a smooth algebraic variety, then TX denotes the
tangentbundleofX.IfY ⊂X isasmoothalgebraicsubvarietythenTY isa
subbundle in TX| . The quotient TX| /TY is called the normal bundle of
Y Y
Y in X, denoted by N X. By taking dual bundles we obtain the cotangent
Y
bundle T∗X and the conormal bundle N∗X. The conormal bundle can be
Y
naturally regarded as a subvariety of T∗X.
Recall that I ⊂ Pn×Pn∗ is the conormal variety and the dual variety
X
X∗ coincides with pr (I ). The projection pr : I0 → X is a projective
2 X 1 X sm
bundle,wherepr ,pr denotetheprojectionsofPn×Pn∗ toitsfactors.More
1 2
precisely,pr identifiesI0 withtheprojectivizationP(N∗ Pn)oftheconor-
1 X Xsm
mal bundle N∗ Pn. Indeed, the choice of a hyperplane H ⊂Pn tangent to
Xsm
X atxisequivalenttothechoiceofahyperplaneT H inthetangentspace
sm x
T Pn, which contains T X. The equation of this hyperplane is an element of
x x
N∗ Pn at x. The Reflexivity Theorem can be reformulated as follows:
Xsm
I =I . (1.1)
X X∗
It is more convenient to prove (1.1) by working with vector spaces instead of
projective spaces.
We assume that Pn =P(V) and Pn∗ =P(V∗). Then we have affine cones
Y = Cone(X) ⊂ V and Y∗ = Cone(X∗) ⊂ V∗. We denote by Lag(Y) the
closure of the conormal bundle N∗ V in the cotangent bundle T∗V.
Ysm
The space T∗V is canonically identified with V × V∗. Denote by pr ,
1
pr the projections of this product to its factors. Then Y∗ coincides with
2
pr (Lag(Y)). Therefore, (1.1) can be reformulated as follows:
2