Topics in Classical Algebraic Geometry

7/29/2019 Topics in Classical Algebraic Geometry

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Topics in Classical Algebraic Geometry

IGOR V. DOLGACHEV and ALESSANDRO VERRA

December 13, 2003


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Contents

9 Apolarity 7

9.1 Apolar schemes . . . . . . . . . . . . . . . . . . . . . . . . . . . 7

9.1.1 The apolar ring of a homogeneous form . . . . . . . . . . 79.1.2 Polar polyhedra . . . . . . . . . . . . . . . . . . . . . . . 9

9.1.3 Generalized polar polyhedra . . . . . . . . . . . . . . . . 11

9.1.4 Secant varieties . . . . . . . . . . . . . . . . . . . . . . . 12

9.1.5 The Waring problems . . . . . . . . . . . . . . . . . . . . 14

9.2 Catalecticant matrices . . . . . . . . . . . . . . . . . . . . . . . . 15

9.3 Examples . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18

9.3.1 Binary forms . . . . . . . . . . . . . . . . . . . . . . . . 18

9.3.2 Quadrics . . . . . . . . . . . . . . . . . . . . . . . . . . 19

9.3.3 Cubic forms . . . . . . . . . . . . . . . . . . . . . . . . . 23

9.4 Plane quartics . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29

9.4.1 Clebsch and Luroth quartics . . . . . . . . . . . . . . . . 299.4.2 The Scorza map . . . . . . . . . . . . . . . . . . . . . . . 34

9.4.3 Duals of homogeneous forms . . . . . . . . . . . . . . . 38

9.4.4 Polar hexagons . . . . . . . . . . . . . . . . . . . . . . . 39

9.4.5 The variety of polar

-gons of a curve of degree

. . 41

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45

10 Cubic surfaces 47

10.1 The

-lattice . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

10.1.1 Lattices . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

10.1.2 The

-lattice . . . . . . . . . . . . . . . . . . . . . . . 49

10.1.3 Roots . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5010.1.4 Exceptional vectors . . . . . . . . . . . . . . . . . . . . . 52

10.1.5 Sixers . . . . . . . . . . . . . . . . . . . . . . . . . . . . 53

10.1.6 Steiner triads of double-sixes . . . . . . . . . . . . . . . . 57

10.1.7 Tritangent trios . . . . . . . . . . . . . . . . . . . . . . . 59

3


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4 CONTENTS

10.1.8 Lines on a nonsingular cubic surface . . . . . . . . . . . . 62

10.1.9 Schurs quadrics . . . . . . . . . . . . . . . . . . . . . . 6410.2 Singularities . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

10.2.1 Non-normal cubic surfaces . . . . . . . . . . . . . . . . . 69

10.2.2 Normal cubic surfaces . . . . . . . . . . . . . . . . . . . 71

10.2.3 Canonical singularities . . . . . . . . . . . . . . . . . . . 71

10.2.4

-nodal cubic surface . . . . . . . . . . . . . . . . . . . . 76

10.2.5 The Table . . . . . . . . . . . . . . . . . . . . . . . . . . 77

10.3 Determinantal equations . . . . . . . . . . . . . . . . . . . . . . 78

10.3.1 Cayley-Salmon equation . . . . . . . . . . . . . . . . . . 78

10.3.2 Hilbert-Burch Theorem . . . . . . . . . . . . . . . . . . . 81

10.3.3 The cubo-cubic Cremona transformation . . . . . . . . . 86

10.3.4 Cubic symmetroids . . . . . . . . . . . . . . . . . . . . . 87

10.4 Representations as sums of cubes . . . . . . . . . . . . . . . . . . 92

10.4.1 Sylvesters pentahedron . . . . . . . . . . . . . . . . . . 92

10.4.2 The Hessian surface . . . . . . . . . . . . . . . . . . . . 95

10.4.3 Cremonas hexahedral equations . . . . . . . . . . . . . . 96

10.5 Automorphisms of a nonsingular cubic surface . . . . . . . . . . 101

10.5.1 Eckardt points . . . . . . . . . . . . . . . . . . . . . . . 101

10.5.2 The Weyl representation . . . . . . . . . . . . . . . . . . 103

10.5.3 Automorphisms of finite order . . . . . . . . . . . . . . . 106

10.5.4 Lefschetz type fixed-point formulas . . . . . . . . . . . . 114

10.5.5 Automorphisms groups . . . . . . . . . . . . . . . . . . . 117

Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 122

11 Geometry of Lines 125

11.1 Grassmanians of lines . . . . . . . . . . . . . . . . . . . . . . . . 125

11.1.1 Tangent and secant varieties . . . . . . . . . . . . . . . . 127

11.1.2 The incidence variety . . . . . . . . . . . . . . . . . . . . 129

11.1.3 Schubert varieties . . . . . . . . . . . . . . . . . . . . . . 135

11.2 Linear complexes of lines . . . . . . . . . . . . . . . . . . . . . . 139

11.2.1 Linear complexes and apolarity . . . . . . . . . . . . . . 141

11.2.2 6 lines . . . . . . . . . . . . . . . . . . . . . . . . . . . . 144

11.2.3 Linear systems of linear complexes . . . . . . . . . . . . 148

11.3 Quadratic complexes . . . . . . . . . . . . . . . . . . . . . . . . 150

11.3.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 15011.3.2 Intersection of 2 quadrics . . . . . . . . . . . . . . . . . . 152

11.3.3 Kummer surface . . . . . . . . . . . . . . . . . . . . . . 154

11.4 Ruled surfaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . 159

11.4.1 Generalities . . . . . . . . . . . . . . . . . . . . . . . . . 159


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CONTENTS 5

11.5 Congruences of lines . . . . . . . . . . . . . . . . . . . . . . . . 161

11.5.1 Class and Order . . . . . . . . . . . . . . . . . . . . . . . 16111.5.2 Congruences of order

: examples . . . . . . . . . . . . . 164

11.5.3 Classification of congruences of order one . . . . . . . . . 167


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6 CONTENTS


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Chapter 9

Apolarity

9.1 Apolar schemes

9.1.1 The apolar ring of a homogeneous form

Let

be a complex vector space of dimension

. Recall from Chapter 1 that

we have a natural polarity pairing

$

&

$ ( 03 2 4

( $ 5 6 $

which extends the canonical pairing

A. By choosing a basis in

and the dual basis in

, we view the ring Sym

as the polynomial algebra

A B D E $ G G G $ D P Rand Sym

as the ring of differential operators

A B T E $ G G G $ T P R. The

polarity pairing is induced by the natural action of operators on polynomials.

Definition 9.1.1. A homogeneous form& Y

is called apolar to a homoge-

neous form(

Y

if

2 4

( b c G

Lemma 9.1.1. For any& Y

$

& f Y

h

and

(

Y

,

24

h

2 4

( b 24 4

h

( G

Proof. By linearity and induction on the degree, it suffices to verify the assertions

in the case when&

b T t

and& f

b T v

. In this case the assertions are obvious.

Corollary 9.1.2. Let (Y

. LetAP

( be the subspace in

spanned

by apolar forms of degree

to(

. Then

AP

( b

x

y

E

2

(

is a homogeneous ideal in the ring Sym

a

.

7


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8 CHAPTER 9. APOLARITY

Definition 9.1.2. The quotient ring

b

Sym

AP

(

is called the apolar ring of(

.

The ring

inherits the grading of Sym

. Since any polynomial& Y

with 5

is apolar to(

, we see that

is killed by the ideaL

b

TE

$ G G G $ TP

. Thus

is an Artinian graded local algebra overA

. Since

the pairing between

and

has values in

E

b A

, we see that

AP

(

is of codimension

in

. Thus

is a vector space of dimen-

sion

overA

and coincides with the socle of

, i.e. the ideal of elements of

annulated by its maximal ideal.

Note that the latter property characterizes Gorenstein graded local Artinianrings, see [Eisenbud, Iarobbino-Kanev].

Proposition 9.1.3. (Macaulay) The correspondence ( 0

is a bijection be-

tween

and graded Artinian quotient algebras Sym

whose socle is

one-dimensional.

Proof. We have only to show how to reconstructA (

from

. The multipli-

cation of5

vectors in

composed with the projection to

defines a linear

map

. Since

a

is one-dimensional. Choosing a basis

, we obtain a linear function on

. It corresponds to an element of

. This is our

(.

Recall that for any closed subscheme

P

is defined by a unique saturated

homogeneous ideal "

inA B D E $ G G G $ D P R

. Its locus of zeroes in the affine space$

P

isomorphic to Spec

A B DE

$ G G G $ DP

R

"

is the affine cone%

"

over

.

Definition 9.1.3. Let (Y

. A subscheme

is called apolar to

(

if its homogeneous ideal "

is contained in AP

( , or, equivalently, Spec

is

a closed subscheme of the affine cone%

"

of

.

This definition agrees with the definition of an apolar homogeneneous form&

.

A homogeneous form& Y

is apolar to(

if and only if the hypersurface

&

is apolar to ( .

Consider the natural pairing

(

b

A

(9.1)


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9.1. APOLAR SCHEMES 9

defined by multiplication of polynomials. The left kernel of this pairing consists

of

& Y

AP

(

such that2

4 4

h

( b c

for all

& f Y

. ByLemma 9.1.1,

2 4 4

h

( b 2 4

h

2 4

( q b c

for all&

fY

. This implies

2 4

( b c

. Thus& Y

AP is zero in AP

(

. This shows that the pairing (11.21) is

a perfect pairing. This is one of the nice features of a Gorenstein artinian algebra.

It follows that the Hilbert poynomial

b

t

E

t

b G G G

is a reciprocal monic polynomial, i.e.

tb

t$

b . It is an important

invariant of a homogenous form(

.

Example 9.1.1. Let ( be the 5 th power of a linear form $Y

. For any

& Y

b

we have

24

$

b 5

5 %

G G G

5 %

$

&

$ b 5 ) $ 2

3&

$ $

where we set

$ 2

t

3

b

)

$

t

G

Here we view& Y

a as a homogeneous function on

. In coordinates,

$ b 5

P

t

E

t

Dt

$

&

b

&

TE

$ G G G $ TP

and

&

$ b

&

E

$ G G G $ P

. Thus we see that

2

(

$ 7 5 $consists of polynomial of degree

vanishing at

$. Assume for

simplicity that$ b D E

. The ideal

2

( is generated by

T

$ T P $ T

E

. The Hilbert

polynomial is equal to

G G G

.

9.1.2 Polar polyhedra

Suppose(

is equal to a sum of powers of nonzero linear forms

( b $

G G G $

9

G

This implies that for any& Y

a ,

24

( b 24

9

t

$

t

b 5 )

9

t

&

$ t $

2

3

t

(9.2)

In particular, taking5 b

, we obtain that

@

$

$ G G G $ $

9 A CE F H P R

b T

& Y

V

&

$t

b c $ b $ G G G $ Y a b

"

$


10/169


where

is the closed subscheme of pointsT B $

R $ G G G $ B $

9

R a

corresponding

to the linear forms$ t

.This implies that the codimension of

@

$

$ G G G $ $

9 A in

is equal to the

dimension of

"

, hence the forms$

$ G G G $ $

9 are linearly independent if and only

if the pointsB $

R $ G G G $ B $

9

R

impose independent conditions on the linear system of

hypersurfaces of degree5

in

.

Suppose(

Y

@

$

$ G G G $ $

9A , then

"

AP

(

. Conversely, if this is true,

we have

(

Y

AP

(

C

"

C

b

@

$

$ G G G $ $

9A

G

If we additionally assume that

"

h

AP

(

for any proper subset

f

of

, we

obtain, after replacing the forms$

f

t

Yby proportional ones, that

( b $

G G G $

9

G

Definition 9.1.4. A polar Y -polyhedron of ( is a set of hyperplanes

tb

$t

$ b

$ G G G $ Y $in

such that

( b $

G G G $

9

$

and, considered as points in

, the hyperplanes

timpose independent condi-

tions in the linear system

H P R

5 .

Note that this definition does not depend on the choice of linear forms defining

the hyperplanes. Nor does it depend on the choice of the equation defining thehypersurface

(

.

The following propositions follow from the above discussion.

Proposition 9.1.4. Let (Y

. Then

tb

$t

Y

$ b $ G G G $ Y $

form a polarY

-polyhedron of(

if and only if the following properties are satisfied

(i)

H P R

5 %

% G G G %

9

AP

( ;

(ii)$

$ G G G $ $

9 are linearly independent in

or

(ii)

H P R

5 %

% G G G %

9

t

AP

(

for any b

$ G G G $ Y

.

Proposition 9.1.5. A set b T

$ G G G $

9

ais a

Yth polar polyhedron of

(

Y

if and only if , considered as a closed subscheme of

, is apolar to

(

but no proper subscheme of

is apolar to(

.


11/169


9.1.3 Generalized polar polyhedra

Proposition 9.1.5 allows one to generalize the definition of a polar polyhedron. We

consider a polar polyhedron a reduced closed subscheme

of

consisting

ofY

points. Obviously,

E

"

b

E

$

"

b Y

. More generally we

may consider non reduced closed subschemes

of

a

of dimensionc

satisfying

E

"

b Y. The set of such subschemes is parametrized by a projective algebraic

variety Hilb9

called the punctual Hilbert scheme of

of lengthY

.

Any

Y

Hilb9

defines the subspace

H P R

5 % b

E

$

"

5

E

$

H P R

5 b

G

The exact sequence

c

E

$

"

5

E

$

H P R

5

E

$

"

$

"

5 c

shows that the dimension of the subspace

@

A

b

E

$

"

5

C

is equal to

E

"

%

"

5 % b Y % %

"

5 GIf

b

b

T

$ G G G $

9

a, then

@

A

b

@

$ G G G $

9

A , where

V

a

is the Veronese map. Hence

@

A

b Y % if the points

$ G G G $

9

are

linearly independent. We say that

is linearly5

-independent if

@

A

b Y % .

Definition 9.1.5. A generalized Y -polyhedron of ( is a linearly 5 -independent sub-

scheme

Y

Hilb9

which is apolar to

(.

Recall that

is apolar to(

if, for each 6 c

,

E

P

$

"

AP

( G(9.3)

In view of this definition a polar polyhedron is a reduced generalized polyhedron.

The following is a generalization of Proposition 9.1.4.

Proposition 9.1.6. A linear independent Y

Hilb9

bb

is a generalized polar

polyhedron of(

Y

if and only if

"

5 AP

( G

Proof. We have to show that the inclusion in the assertion implies

"

5

AP

(

for any 7 5

. For any& f Y

and any& Y

, the product& & f

be-

longs to

. Thus2

4 4

h

( b c. By the duality,

2 4

( b c, i.e.

& Y

AP

(

.


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Example 9.1.2. Let b

G G G

Y

Hilb9

be the union of fat

points

, i.e. at each t

Y

the ideal

"

is equal to the t

th power of themaximal ideal. Obviously,

Y b

t

t

%

t

%

G

Then

H P R

5 % consists of hypersurfaces of degree which have singularity

at t

of multiplicity6 t

for each b

$ G G G $

. One can show (see [Iarrobino-

Kanev], Theorem 5.3 ) that

is apolar to(

if and only if

( b $

G G G $

$

where

tb B $

tR

and

t

is a homogeneous polynomial of degree

t%

.

Remark 9.1.1. It is not known whether the set of generalizedY

-polyhedra of(

is

a closed subset of Hilb9

a . It is known to be true for

Y 7 5 since in this

case

b

V b

% Yfor all

Y

Hilb9

(see [IK], p.48).

This defines a regular map of Hilb9

to the Grassmannian

$

and

the set of generalizedY

-polyhedrons is equal to the preimage of a closed subset

of subspaces contained in AP

(

. Also we see that

"

5 q b c

, hence

is

always linearly5

-independent.

9.1.4 Secant varieties

The notion of a polar polyhedron has a simple geometric interpretation. Let

V

$ $ 0 $

$

be the Veronese map. Denote by VerP

its image. Then(

Y

T c arepresents

a pointB ( R

in

. A set of hyperplanes

t b

$ t $ b $ G G G $ Y $repre-

sents a set of pointsB $

t

Rin the Veronese variety Ver

P

. It is a polarY

th polyhedron

of(

if and only ifB ( R

belongs to the linear span@

B $

R $ G G G $ B $

9

R

A , a

Y % -secant

of the Veronese variety.

Recall that for any irreducible nondegenerate projective variety

of

dimension

its

-secant variety Sec

is defined to be the Zariski closure of the

set of points in

which lie in the linear span of dimension

of some set of

linear independent points in

.

Counting constants easily gives

Sec

7

!

%

$ G


13/169


The subvariety

is called

-defective if the inequality is strict. An example

of a

-defective variety is a Veronese surface in

.A fundamental result about secant varieties is the following lemma whose mod-

ern proof can be found, for example in [Dale], [Zak,Prop.1.10].

Lemma 9.1.7. (A. Terracini) Let $ G G G $ be general

points in

and

be a general point in their span. Then

PT

Sec

b

@

PT

$ G G G $

PT

A

$

where PT

denotes the embedded Zariski tangent space of a closed subvariety

of a projective space at a point .

The inclusion part

@

PT

$ G G G $ PT

A

PT

Sec

is easy to prove. We assume for simplicity that

b . Then Sec

con-

tains the cone%

$ which is sweeped by the lines

@

$

A

$

Y

. Therefore

PT

%

$ PT

Sec

. However, it is easy to see that PT

%

$

contains PT

.

Corollary 9.1.8. Sec

b

if and only if for any

general points of

there exists a hyperplane section of

singular at these points. In particular, if

7

% , the variety

is

-defective if and only if for any

general points of

there exists a hyperplane section of

singular at these points.


Ver

P

P

P

be the image of

P

under a Veronesemap defined by homogeneous polynomials of degree

5. Assume

6

P

P

% . A hyperplane section of

is isomorphic to a hypersurface of degree

5

in

P

. Thus Sec

VerP

b

if and only if for any

general points in

P

there exists a hypersurface of degree5

singular at these points.

Take b

. Then b 5

and 7

% b

for

6

5 %

.

Since

5

there are no homogeneous forms of degree5

which have

multiple roots. Thus the Veronese curve

b

is not

-degenerate for

6

5 %

.

Take b

and5 b

. For any two points in

there exists a conic singular

at these points, namely the double line through the points. This explains why a

Veronese surface

is

-defective.Another example is Ver

and

b

. The expected dimension of

Sec

is equal to

. For any 5 points in

there exists a conic passing through

these points. Taking it with multiplicity 2 we obtain a quartic which is singular at

these points. This shows that Ver

is

-defective.


14/169


The following corollary of Terracinis Lemma is called the First main theorem

on apolarity in [Ehrenborg-Rota]. They gave an algebraic proof of this theoremwithout using (or probably without knowing) Terracinis Lemma.

Corollary 9.1.9. A general form(

Y

admits a polar

Yth polyhedron if

and only if there exists linear forms$

$ G G G $ $

9

Y

such that for any nonzero& Y

a the ideal

2

&

does not contain

T $

$ G G G $ $

9

a.

Proof. A general form(

Y

admits a polar

Yth polyhedron if and only

if the secant variety Sec 9

VerP

is equal to the whole space. This means that for

some pointsB $

R $ G G G $ B $

9

Rthe span of the tangent spaces at the points

@

B $

R $ G G G $ B $

9

R

A

is equal to the whole space. By Terracinis Lemma, this is equivalent to that the

tangent spaces of the Veronese variety at the pointsB $

t

Rare not contained in a

hyperplane defined by some

& Y

b

. It remains to use that thetangent space of the Veronese variety atB $ t R

is equal to the projective space of

all homogeneous forms of the form$

t

$ $ $

Y

(see Exercises). Thus, for any

nonzero& Y

, it is impossible that2

F

&

b c

for all$

and

. But

2

F

&

b c

for all$

if and only if2

F

&

b c

. This proves the assertion.

The following fundamental result is due to J. Alexander and A. Hirschowitz.

Theorem 9.1.10. VerP

is

-defective if and only if

$ 5 $

b

$ $ $

$ $ $

$ $ $

$

$ $

$ $

G

In all these cases the secant variety Sec

VerP

is a hypersurface.

For the sufficiency of the condition, only the case

$

$ is not trivial. It

asserts that for general points in

there exists a cubic hypersurface which is

singular at these points. Other cases are easy. We have seen already the first two

cases. The third case follows from the existence of a quadric through 9 general

points in

. The square of its equation defines a quartic with 9 points. The last

case is similar. For any 14 general points there exists a quadric in

containing

these points.

Corollary 9.1.11. AssumeY

6

P

P

. Then a general homogeneous poly-

nomial(

Y

A B DE

$ G G G $ DP

R

can be written as a sum of5

th powers ofY

linear forms

unless

$ 5 $ Y b

$ $ $

$ $ $

$ $ $

$

$ $

$ $ .

9.1.5 The Waring problems

The well-known Waring problem in number theory asks about the smallest number

Y

5

such that each natural number can be written as a sum ofY

5 5

th powers of


15/169

9.2. CATALECTICANT MATRICES 15

natural numbers. It also asks in how many ways it can be done. Its polynomial

analog asks about the smallest numberY

5 $

such that a general homogeneouspolynomial of degree

5

in

variables can be written as a sum ofY 5

th powers

of linear forms.

The Alexander-Hirschowitz Theorem completely solves this problem. We have

Y

5 $

is equal to the smallest natural numberY E

such thatY E

6

P

P

unless

$ 5 b

$ $

$ $

$ $

$

$

$

, whereY

5 $ b Y E

.

Other versions of the Waring problem ask the following questions:

(W1) Given a homogeneous form(

Y

A B DE

$ G G G $ DP

R, study the subvari-

ety

( Y

of

P

H

9

R

which consists of polarY

-polyhedra of(

or more

general the subvariety

( Y of Hilb

9

P

parametrizing generalized

Y-

polyhedra.

(W2) For givenY

find the equations of the closure PS

Y $ 5 in

A B DE

$ G G G $ DP

R

of the locus of homogeneous forms of degree5

which can be written as a sum

ofY

powers of linear forms.

Note that PS

Y $ 5 is the affine cone over the secant variety Sec 9

VerP

.

In the language of secant varieties, the variety

( Y

is the set of linear

independent sets ofY

points

$ G G G $

9 in VerP

such thatB ( R

Y

@

$ G G G $

9

A . The

variety

( $ Y

is the set of linearly independent

Y

Hilb9

such that

B ( R

Y

@

A . Note that we have a natural map

( $ Y

Y $

$ 0

@

A

$

where

Y $

is the Grassmannian of Y -dimensional subspaces of

.This map is not injective in general.

Also note that

( $ Y

embeds naturally in

by assigning to

T $

$ G G G $ $

9

a

the product$

$

9 . Thus we can compactify

( $ Y

by taking its closure in

. In general this closure is not isomorphic to

( $ Y

.

9.2 Catalecticant matrices

Let(

Y

. Consider the linear map

ap V

$

&

0 2 4

( G(9.4)

Its kernel is the space of forms of degree

which are apolar to(

.

By the polarity duality, the dual space of

can be identified with

. Applying Lemma 9.1.1, we obtain

ap

b

ap

G

(9.5)


16/169


Assume that( b

5

9

t

$

t

for some$

t

Y

. It follows from (9.2) that

ap

@

$

$ G G G $ $

9 A

$

and hence

rank

ap

7 Y G

(9.6)

If we choose a basis in

and a basis in

, then ap

is given by a matrix of size

P

P

whose entries are linear forms in coefficients of(

.

Choose a basis

E$ G G G $

P

in

and the dual basisD

E$ G G G $ D

P

in

. Consider

a monomial basis in

(resp. in

which is lexigraphically ordered.

The matrix of ap

with respect to these bases is called the

th catalecticant matrix

of(

and is denoted by Cat

( . Its entries

are parametrized by pairs

$

Y

P

P

with

b 5 % and

b . If we write

( b 5 )

)

D

$

then

b

G

This follows easily from the formula

T

t

E

T

t

P

D

v

E

D

v

P

b

! " #

H

"

R

# if$ %

6 c

cotherwise

G

Considering

as independent variablesD

, we obtain the definition of a general

catalecticant matrix Cat

5 $

.Example 9.2.1. Let

b . Write

( b 5

t

E

t

t D

t

E

D

t

. Then

Cat

( b %'

'

'(

E

G G G

G G G

......

G G G

...

G G G

) 0

0

0

1

A matrix of this type is called a Hankel matrix. It follows from (9.6) that(

Y

PS

Y $ 5

implies that allY Y

minors of Cat

(

are equal to zero.

Thus we obtain that Sec 9

Ver

is contained in the subvariety of

defined by

Y

Y -minors of the matrices

Cat

5 $ b %'

'

'(

D E D

G G G D

D

D

G G G D

......

G G G

...

D

D

G G G D

) 0

0

0

1

$ b $ G G G $

!

5 % Y $ Y


17/169

9.2. CATALECTICANT MATRICES 17

For example, ifY b

, we obtain that the Veronese curve Ver

satisfies

the equationsDF t D v % D

D

b c

, where b $

. It is well known that theseequations generate the homogeneous ideal of the Veronese curve.

Assume5 b

. Then the Hankel matrix is a square matrix of size

. Its

determinant vanishes if and only if(

admits a nonzero apolar form of degree

.

The set of such(

s is a hypersurface inAC B D E $ D

R

. It contains a Zariski open

subset of forms which can be written as a sum of

powers of linear forms (see

section 9.3.1).

For example, take b

. Then the equation

%

(

E

)

1

b c(9.7)

describes binary quartics

( b E

DE

D

E

D

D

E

D

DE

D

D

which lie in the Zariski closure of the locus of quartics represented in the form

ED

E

ED

DE

D

. Note that a quartic of this form has simple

roots unless it has a root of multiplicity 4. Thus any binary quartic with simple

roots satisfying equation (9.7) can be represented as a sum of two powers of linear

forms.

The cubic hypersurface in

defined by equation (9.7) is equal to the 1-secant

variety of a Veronese curve in

.

Recall that each(

Y

defines its apolar ring

bd A B DE

$ G G G $ DP

R

AP

(

.Let

b

t

E

t

t

be its Hilbert polynomial. Note that

AP

t

( b

Ker

apt

b

P

t

%rankCat

t

( G

Therefore,

t brankCat

t

( $

and

b

t

E

rankCat t

(

t

G (9.8)

It follows from (9.5) that

rankCatt

( b

rankCat t

(


18/169


confirming that

is a reciprocal monic polynomial.

Suppose5 b

is even. Then the coefficient at

in

is equal torankCat

(

. The matrix Cat

(

is a square matrix of size

P

. One can show

that for a general(

, this matrix is nonsingular. A polynomial(

is called degener-

ate if

Cat

( b c. Thus, the set of degenerate polynomials is a hypersurface

given by the equation

Cat

$ b c G

(9.9)

The polynomial in variablesD

$

b 5is called the catalecticant determinant.

Example 9.2.2. Let5 b

. It is easy to see that the catalecticant polynomial is the

discriminant polynomial. Thus a quadratic form is degenerate if and only if it is

degenerate in the usual sense. The Hilbert polynomial of a quadratic form(

is

b

$

where

is the rank of the quadratic form.

Example 9.2.3. Suppose( b D

E

G G G D

9

$ Y 7 . Then

D

t

E

$ G G G $ D

t

9 are linearly

independent for any

, and hence rankCatt

( b Y

forc

5

. This shows that

b Y

G G G

G

Let be the set of reciprocal monic polynomials of degree5

. One can stratify

the space

by setting, for any

2

Y

,

b T (

Y

V

b 2 a G

If(

Y

PS

Y $ 5

we know that

rankCat

( 7

Y $ 5 $

b

!

Y $

P

P

$

P

P

G

One can show that for general enough(

, we have the equality (see [Iarrobono-

Kanev, Lemma 1.7]). Thus there is a Zariski open subset of PS

Y $ 5 which

belongs to the strata

, where

2 b5

t

E

Y $ 5 $ t

t

G

9.3 Examples

9.3.1 Binary forms

This is the case b

. The zero subscheme of a homogeneous form of de-

gree5

in 2 variables(

D E $ D

is a positive divisor

b5

t t

of degree5

.

Each such divisor is obtained in this way. Thus we can identify

with

H P R

5

(

b

and also with the symmetric product

a

H

R

b

a

and


19/169

9.3. EXAMPLES 19

the Hilbert scheme Hilb

. A generalized

Y-polyhedron of

(is a posi-

tive divisor b

5

t

t B $ t R

of degreeY

in

such thatB ( R

Y

@

A

b

E

$

"

5

C

. Note that in our case

is automatically linearly inde-

pendent (because

"

5 b c

). Obviously,

E

$

"

5

consists of poly-

nomials of degree5

which are divisible by&

b

, where

t

Y

AP

$t

.

In coordinates, if$ t b t D E t D

, then

t b t T E % t T

. Thus(

is orthogonal

to this space if and only if2 4 4

h

( b c

for all& f Y

9

. By the apolarity

duality this means that2 4

( b c, hence

& Y

AP

(

9 . Thus we obtain

Theorem 9.3.1. A positive divisor

b

$

$

of degree is a generalized

Y-polyhedron of

(if and only if

Y

AP

(

9 .

Corollary 9.3.2. Assume b . Then

( Y b

AP 9

( G

Note that the kernel of the map

9

9

$

&

03 2 p 4

(

is of dimension6

9

%

9

b Y %

5 % Y b Y % 5

.

Thus2 4

( b c

for some nonzero& Y

9

, whenever Y 5

. This shows

that a(

has always generilized polarY

th polyhedron forY 5

. If5

is even,

a binary form has an apolar5

-form if and only

Cat

( b c. This is a

divisor in the space of all binary5

-forms.

Example 9.3.1. Take5 b

. Assume that(

admits a polar 2-polyhedron. Then

( b

DE

D

DE

D

G

It is clear that(

has 3 distinct roots. Thus, if( b

DE

D

DE

D %

has a double root, it does not admit a polar

-polyhedron. However, it admits

a generalized

-polyhedron defined by the divisor

, where b

$ %

. In

the secant variety interpretation, we know that any point in

either lies

on a unique secant or on a unique tangent line of the Veronese cubic curve. The

space AP

(

is always one-dimensional. It is generated either by a binary quadric

%

E

%

E

or by

%

E

.

Thus

(

consists of one point or empty but

( always consist of

one point. This example shows that

(

b

(

in general.

9.3.2 Quadrics

It follows from Example 9.1.1 that Sec

P

b

if only if there exists

a quadric with

singular points in general position. Since the singular locus


20/169


of a quadric

is a linear subspace of dimension equal to corank

% , we

obtain that SecP

Ver

P

b

, hence any general quadratic form canbe written as a sum of

squares of linear forms$ E $ G G G $ $ P

. Of course, linear

algebra gives more. Any quadratic form of rank

can be reduced to sum of

squares of the coordinate functions. Thus we may assume that

b D

E

G G G D

P

.

Suppose we also have

b $

E

G G G $

P

. Then the linear transformationD t 0 $ t

preserves

and hence is an orthogonal transformation. Since polar polyhedra of

and

are the same, we see that the projective orthogonal group PO

acts transitively on the set

(

of polar

th polyhedra of

. The

stabilizer group

of the coordinate polar polyhedron is generated by permutations

of coordinates and diagonal orthogonal matrices. It is isomorphic to the semi-direct

product

P

P

(the Weyl group of roots systems of type

P$

P

), where we use

the standard notation

for the 2-elementary abelian group

. Thus we

obtain

Theorem 9.3.3. Let

be a nondegenerate quadratic form in

variables. Then

(

b

PO

P

P

G

The dimension of

is equal to

.

Example 9.3.2. Take b

. Using the Veronese map

V

, we con-

sider a nonsingular quadric

as a point

in

not lying on the conic% b

DE

D

% D

. A polar 2-gon of

is a pair of distinct points

$

on%

such

that

Y

@

$

A

. It can be identified with the pencil of lines through

with thetwo tangent lines to%

deleted. Thus

$

b

T c $ a b A

. There

are two generalized 2-gons

E

and

x defined by the tangent lines. Each of

them gives the representation of

as$

$

, where

$ t

are the tangents. We have

( b

(

(

b

.

It is an interesting question to define a good compactification of this space

similar to the one we found in Chapter 2 in the case b

.

Let Y

be a non-degenerate quadratic form. EachT $

$ G G G $ $ P

a

Y

defines a

-dimensional subspace in

b

@

A

(

b

A

equal to@

$

$ G G G $ $

P

A

@

A . This defines a map

V

$ G

(9.10)

This map is injective. In fact, suppose@

$

$ G G G $ $

P

A

b

@

$ G G G $

P

A

@

(

A .

Then

$

b

G G G P

P

(


21/169

9.3. EXAMPLES 21

for some scalars

$ G G G $ P

$ . Since

(is a linear combination of

t

, we may

assume that b c

. But$

, being of rank 1, cannot be equal to a sum of

linearindependent squares of linear forms unless

b

. Thus we may assume that

$

t

b t

t

for some

. This immediately implies(

is a sum of

squares on linear

forms contradicting the assumption that(

is of rank

.

Now consider the dual quadratic form

Y

a

of

considered as a linear

function on

. For any two quadratic forms

$

Y

we can consider

the matrix

E

G G G

P

E

G G G P

$(9.11)

where

t$

t

are partial derivatives in the variableD

t

. Let

t v$ c 7

7 ,

be the

minors of the matrix. The function t v V

$ 0

v

is a

skew-symmetric bilinear form on the space

. Of course, to define the partial

derivatives we need to choose a basis in

. However, a change of a basis multiplies

the matrix (9.11) by an invertible constant matrix, and hence t v

is unchanged up

to a nozero scalar factor. Thus, to be more precise,

t v

Y

G

I claim that

t v

$

b cfor any

Y

. In fact, choose a basis in

such that

b 5

P

t

E

D

t

. Then

b 5

P

t

E

T

t

. By linearity, we may assume that

b Dt

Dv

$ 7 .

$

b

P

t

E

T

t

D

T

Dt

Dv

% D

T

Dt

Dv

G

If

b $ $

b $ we get zero. If

b $ $

b , we get

$

b

P

t

E

T

t

% D

Dv

b c G

Finally, if b $ $ b

, we get

$

b

P

t

E

T

t

D

t

% D

v

b c G

Since

belongs to the kernel of the bilinear form

t v

we can consider each function

t vas a skew Recall that the Grassmann variety

$ carries the natural rank

vector bundle

, the tautological bundle. Its fibre over a point

Y

$ is

equal to

. It is a subbundle of the trivial bundle

H

R associated to the vector

space

. We have a natural exact sequence

c

H

R

c $


22/169


where

is the universal quotient bundle, whose fibre over

is equal to

.

We can consider each element

of

as a section of the trivial bundle

H

R

. Restricting

to the subbundle

, we get a section of the vector

bundle

. Thus we can view our functions t v

as sections of

defined

up to a constant, i.e. elements of the projective space

E

$ $

.

Lemma 9.3.4. The image of the map (9.10) is contained in the set of common zeros

of the sections

t v

.

Proof. We have to show that t v

is identical to zero on each subspace@

$

$ G G G $ $

P

A ,

containing

. Again, without loss of generality, we may assume that

b 5 D

9

$

b

5

T

9 . By linearity, it suffices to show that

$

t

$ $

v

b c

for allc 7

$ 7

$ 7

7 . We have

$

t

$ $

v

b

P

9

E

T

9

T

$

t

T

$

v

% T

$

v

T

$

t

b

P

9

E

T

9

$t

$v

T

$t

T

$v

% T

$v

T

$t

G

Let$

tb 5

P

9

E

9

D

9

$ $v

b 5

P

9

E

9

D

9 . Since5

P

9

E

$

9

b 5

P

9

E

D

9 , we easily

see, that the coefficients of the linear forms$

9 , considered as vectors inA

P

, are

orthogonal with respect to the dot-product. Therefore

P

9

E

T

9

$ t $ v b

P

9

E

9

9

b c G

This proves the assertion.

The next result has been proven already, by different method, in Chapter 2, Part

I.

Corollary 9.3.5. Assume b . Then the image of

in

$ , embed-

ded in the Plucker space

is an open Zariski subset of the intersection

of

$ with a linear space of codimension 3.

Proof. We have

b

, so

$

(

b

$

is of dimension

. Hyper-

planes in the Plucker space are elements of the space

. Note that thefunctions

t v

are linearly independent. In fact, assume

b5

D

t

. if we take

b D E D

D

$ b % D

E

D

D

, we see that E

$

b c $

$ b

c $ E

$ b c. Thus a linear dependence between the functions

t v

implies

the linear dependence between two of the functions. It is easy to see that no two


23/169

9.3. EXAMPLES 23

functions are proportional. So our 3 functions

t v$ c 7

7 span a codi-

mension 3 subspace

in the Plucker space. The line bundle

H

R

is equalto

. By the previous lemma, the image of

is contained in the in-

tersection

$

. This is a 3-dimensional subvariety of

$

, and hence

contains

as an open Zariski subset.

In Chapter 2 we proved also that

is a smooth Fano 3-fold of degree 5. I do

not know how to prove the smoothness of

using the present method. Note that in

Chapter 2, we identified each point of

with a closedc

-dimensional subscheme

of the Veronese surface Ver

such thatB

R

Y

@

A . Using our interpretation of

generalized polar polyhedra, we see that b

.

If

, the vector bundle

is of rank

b

P

. The zero locus of

its nonzero section is of expected codimension equal to

. We have

P

sections t v

and

$ a b

P

% % . For example, when

b

, we have

6 sections

t v

each vanishing on a codimension 3 subvariety of 18-dimensional

Grassmannian

$ . So there must be some dependence between the functions

t v.

Remark 9.3.1. One can also consider the varieties

Y for

Y . For

example, we have

D

E

b

DE

D

DE

% D

% D

$

D

E

b

DE

D

DE

% D

%

D

D

D

% D

G

This shows that

$

are not empty for any nondegeneratequadric

. I do not know anything about their structure.

9.3.3 Cubic forms

We will start with cubic forms in 3 variables. Since for any three general points in

there exists a plane cubic singular at these points (the union of three lines), a

general ternary cubic form does not admit polar triangles. Of course this is easy to

see by counting constants.

A plane cubic curve projectively isomorphic to the cubic

D

E

D

D

will

be called a Fermat cubic. Obviously such a curve admits a non-degenerate polar

3-polyhedron.

Theorem 9.3.6. A plane cubic admits a polar 3-polyhedron if and only if either it

is a Fermat cubic member or it is equal to the union of three distinct concurrent

lines.


24/169


Proof. Suppose( b $

$

$

. Without loss of generality we may assume that

$

is not proportional to$

. Thus, after coordinate change( b D

E

D

$

. If$

D E $ D

$ D

does not depend onD

, the curve

(

is the union of three distinct

concurrent lines. Otherwise we can change coordinates to assume that$ b D

and

get a Fermat cubic.

Remark 9.3.2. If(

is a Fermat cubic, then its polar 3-polyhedron is unique. Its

sides are the three first polars of(

which are double lines.

By counting constants, we see that a general cubic admits polar quadrangles.

We call a polar quadrangleT

$ G G G $

anondegenerate if it is defined by 4 points

in

a

no three of which are collinear. It is clear that a polar quadrangle is non-

degenerate if and only if the linear system of conics in

through the points

B $

R $ G G G $ B $

R

is an irreducible pencil (i.e. a linear system of dimension 1 whose

general member is irreducible). This allows us to define a degenerate generalized

4-polyhedron of(

as a generalized polyhedron

of(

such that

"

is an

irreducible pencil.

Lemma 9.3.7.(

admits a degenerate polar 4-polyhedron if and only if

( is

one of the following curves:

(i) a Fermat cubic;

(ii) a cuspidal cubic;

(ii) the union of three concurrent lines (not necessary distinct);

Proof. We have

D

E

D

D

b

DE

D

DE

D

DE

D

D

$

where b

t

.

We have

DE

D

D

E

D

b

DE

D

DE

D

% D

E

%

D

G

Since the union of three distinct concurrent lines

( is projectively equivalent

to

D E D

D E

D

, we see that(

admits a degenerate quadrangle.

We also have

DE

D

b

DE

D

DE

% D

%

D

E

D

$

D

E

b

D E D

D E

D

%

% D E 5 D

%

% 5

D

$


25/169

9.3. EXAMPLES 25

where

b

5 $ b 5

$

b c. This shows that case (iii) occurs.

All cuspidal cubics are projectively equivalently. So it is enough to demonstratea degenerate 4-polyhedron for

D

E

D

D

. We have

D

E

D

D

b

D

D

D

% D

% D

D

E

G

Now let us prove the converse. Suppose

( b $

$

$

$

$

where$

$ $

$ $

vanish at a common point

. Let

be a linear form on

corre-

sponding to

. We have

2

( b $

$

$

$

$

$

$

$

b $

$

G

This shows that the first polar2

( b 2

( is either the whole

or a double

line

$

. In the first case

(is the union of three concurrent lines. Assume

the second case happens. We can choose coordinates such that b

$ c $ c and

$ b

DE

. Write

( b E

D

E

D

E

DE

$

where t

are homogeneous forms of degree

in variablesD

$ D

. Then2

( b

T E

( b

D

E

E D E

. This can be proportional toD

E

only if

b

b

c

. Thus

( b

D

E

D

$ D

. If

does not multiple linear factors, we

can choose coordinates such that

b D

D

, and get the cubic. If

has a

linear factor with multiplicity 2, we reduce

to the formD

D

. This is the case

of a cuspidal cubic. Finally, if

is a cube of a linear form, we reduce the latter to

the formD

and get three concurrent lines.

Remark 9.3.3. The set of Fermat cubics is a hypersurface in the space

isomorphic to the homogeneous space PSL

G

. Its closure in

consists of curves listed in the assertion of the previous lemma and also reducible

cubics equal to the unions of irreducible conics with its tangent line.

Lemma 9.3.8. The following properties equivalent

(i) AP

(

b T c a ;

(ii)

AP

(

;

(iii)

( is equal to the union of three concurrent lines.


26/169


Proof. By the duality

(

b

A, we have

b

%

2

( b

b

%

AP

( G

Thus

AP

( b

2

(

. This proves the equivalence of (i) and (ii).

By definition, AP

(

b T c a

if and only if2

4

( b c

for some nonzero linear

operator 5

tT

t

. After a linear change of variables, we may assume that&

b TE

,

and thenT E

( b c

if and only if(

does not depend onD E

, i.e.

(

is the union

of three concurrent lines.

Lemma 9.3.9. Let be a nondegenerate generalized 4-polyhedron of ( . Then

"

is a pencil contained in

AP

(

. Conversely, let

be a 0-dimensional

cycle of degree 4 in

. Assume that

"

is a pencil without fixed part

contained in AP

( . Then is a nondegenerate generalized 4-polyhedron of(

.

Proof. The first assertion follows from the definition of non-degeneracy and Propo-

sition 9.1.6. Let us prove the converse. Let

be the pencil of con-

ics

"

. Since AP

( is an ideal, the linear system

of cubics of the form

, where

$

are linear forms, is contained in

AP

( .

Obviously it is contained in

"

. Since

"

has no fixed part we may

choose

and

with no common factors. Then the map

"

defined by

$

is injective hence

b

. Assume

"

6

. Choose 3 points in general position on an irreducible member%

of

"

and 3 non-collinear points outside%

. Then find a cubic

from

"

which passes through these points. Then

intersects%

with total multiplicity

b , hence contains

%. The other component of

must be a line pass-

ing through 3 non-collinear points which is absurd. So,

"

b and we

have b

"

. Thus

"

AP

( and, by Proposition 9.1.6,

is a

generalized 4-polyhedron of(

.

Corollary 9.3.10. Suppose(

is not the union of three concurrent lines. The subset

of

( $

consisting of nondegenerated generalized 4-polyhedrons is isomorphic

to an open subset of the plane

AP

(

consisting of pencils with no fixed part.

Example 9.3.3. Let

(

be the union of an irreducible conic and its tangent line.After a linear change of variables we may assume that

( b D E

D E D

D

. It

is easy to check that AP

(

is spanned byT

$ T

T

$ T

% T E T

. It follows from

Lemma 9.3.7 that(

does not admit degenerate polar 4-polyhedra. Thus any polar

4-polyhedra of(

is the base locus of an irreducible pencil in

AP

(

. However,


27/169

9.3. EXAMPLES 27

it is easy to see that all nonsingular conics inAP

( are tangent at the point

c $ $ c

. Thus no pencil has 4 distinct base points. This shows that

(

b G

Of course,

( $

b GAny irreducible pencil in

AP

( defines a generalized

4-polyhedron. It is easy to see that the only reducible pencil is

T

T

T

.

Thus

( $ contains a subvariety isomorphic to a complement of one point in

b AP

(

. To compactify it by

we need to find one more generalized

-polyhedron. Consider the subscheme

of degree 4 concentrated at the point

$ c $ c with ideal at this point generated by

$

$ , where we use inhomo-

geneous coordinates b T

TE

$ b T

TE

. The linear system

"

is of

dimension 5 and consists of cubics of the form

TE

T

T

T

T

$ T

.

Thus is linearly 3-independent. One easily computes AP

( . It is generated byall monomials except

T

T

andT

ET

and also the polynomialT

ET

% T

T

. We see

that

"

AP

(

. Thus

is a generalized 4-polyhedron of(

. It is non-

degenerate since

"

is the pencil

T

T

T

. So, we see that

( $

is isomorphic to the planeAP

(

.

Example 9.3.4. Let

( be an irreducible nodal cubic. Without loss of generality

we may assume that( b D

DE

D

D

DE

. The space of apolar quadratic forms

is spanned byT

E

$ T

T

$ T

% T

. The netAP

( is base-point-free. It is easy to

see that its discriminant curve is the union of three distinct non-concurrent lines.

Each line defines a pencil with singular general member but without fixed part. So,

( $ b AP

(

.

Example 9.3.5. Let

( be the union of an irreducible conic and a line which in-

tersects the conic transversally. Without loss of generality we may assume that( b

D E

D

E

D

D

. The space of apolar quadratic forms is spanned byT

$ T

$ T

T

%

T

E

. The netAP

( is base-point-free. It is easy to see that its discriminant

curve is the union of a conic and a line intersecting the conic transversally. The

line defines a pencil with singular general member but without fixed part. So,

( $ b AP

(

.

Example 9.3.6. Let

( be a cuspidal cubic. Without loss of generality we may

assume that( b D

D E D

. The space of apolar quadratic forms is spanned by

T

E

$ TE

T

$ T

T

. The net

AP

(

has 2 base points

c $ $ c

and

c $ c $

. The point

c $ c $

is a simple base-point. The point

c $ $ c

is of multiplicity 2 with the ideallocally defined by

$

. Thus base-point scheme of any irreducible pencil is not

reduced. There are no polar 4-polyhedra defined by the base-locus of a pencil of

conics inAP

( . The discriminant curve is the inion of two lines, each defining

a pencil with a fixed line. So

AP

(

minus 2 points parametrizes generalized


28/169


polar 4-polyhedra. We know that

( admits degenerate polar 4-polyhedra. Thus

( $

is not empty and consists of degenerate polar 4-polyhedra.Example 9.3.7. Let

(

be a nonsingular cubic curve. We know that its equation

can be reduced to a Hesse form

D

E

D

D

D E D

D

, where

b c

.

The space of apolar quadratic forms is spanned by T

ET

% T

$ T

T

% T

E

$ TE

T

%

T

. The curve

(

is Fermat if and only if

% b c

. In this case the net has

3 ordinary base points and the discriminant curve is the union of 3 non-concurrent

lines. The net has 3 pencils with fixed part defined by these lines. Thus the set of

nondegenerate generalized polyhedrons is equal to the complement of 3 points in

AP

(

. We know that a Fermat cubic admits degenerate polar 4-polyhedra.

Suppose

(

is not a Fermat cubic. Then the net

AP

(

is base-point-free.

Its discriminant curve is a nonsingular cubic. All pencils are irreducible. There are

no degenerate generalized polygons. So,

( $ b

AP

(

.Example 9.3.8. Assume that

( b

DE

D

D

is the union of 3 non-concurrent

lines. Then AP

(

is spanned byT

E

$ T

$ T

. The net

AP

(

is base-point-

free. The discriminant curve is the union of three non-concurrent lines representing

pencils without fixed point but with singular general member. Thus

( $ b

AP

(

.

It follows from the previous examples that AP

( is base-point-free net of

conics if and only if(

does not belong to the closure of the orbit of Fermat cubics.

Theorem 9.3.11. Assume that ( does not belong to the closure of the orbit of

Fermat cubics. ThenAP

( is base-point-free net of conics and

( $

(

b

AP

(

(

b

G

The variety

( $

is isomorphic to the open subset ofAP

(

whose com-

plement is the curve

of pencils with non-reduced base-locus. The curve

is a

plane sextic with 9 cusps if

( is a nonsingular curve, the union of three non-

concurrent lines if

( is an irreducible nodal curve or the union of three lines,

and the union of a conic and its two tangent lines if

( is the union of a conic

and a line.

Proof. The first assertion follows from the Examples 9.3.4-9.3.8. Since the linear

system of conicsAP

( is base-point-free, it defines a regular map

V

a AP

(

G

The pre-image of a line is a conic from

. The lines through a point

in

AP

(

define a pencil with base locus

. Thus pencils with non-reduced locus are

parametrized by the branch curve

of the map

.


29/169


30/169


of constant suggests that this is possible. This remarkable fact was first discovered

by J. Luroth in 1868. Suppose a quartic admits a polar pentagonT $

$ G G G $ $

a

. Let% b

be a conic in

passing through the pointsB $

R $ G G G $ B $

R

. Then Y

AP

(

. The space AP

(

b T c a

if and only

Cat

( b c

. Thus the

set of quartics admitting a polar pentagon is the locus of the catalecticant invariant

on the space

. It is a polynomial of degree 6 in the coefficients of a

homogeneous form of degree 4.

Definition 9.4.1. A plane quartic admitting a polar pentagon is called a Clebsch

quartic.

A Clebsch quartic is called nondegenerate if

AP

( b . Thus the polar

pentagon of a nondegenerate Clebsch quartic lies on a unique conic. We call it the

associated conic. The associated conic is reducible if and only if the correspondingoperator is the product of two linear operators. This means that the second polar

2

( b c

for some points $

Y

.

Proposition 9.4.1. Let (Y

be such that the second polar

2

( b cfor

some $

Y

a . Then, in appropriate coordinate system

( b

D E $ D

D E

D

$ D

$

b

$

( b

D

$ D

D E

D

$ D

$ b

G

In particular,2

( b cif and only if

( has a triple point.

Proof. Suppose

b

. Choose coordinates such that b

$ c $ c $

b

c $ c $

and write

( b

t

E

t

D

$ D

D

t

E

G

Then2 b

t

$ 2

( b

( b c. Now the assertions easily follow.

We will assume that the apolar conic of a nondegenerate Clebsch quartic is

irreducible.

LetT B $

R $ G G G $ B $

R a

be a polar pentagon of(

such that( b $

G G G $

. For

any 7

7 , let

t v

b

$t

$v

Y

. We can identify

t v

with

a linear operator&

t v

Y

(defined up to a constant factor). Obviously, 2 4

(

coincides with the first polar2

(

. Applying&

t v

we obtain

2 4

( b 2 4

$

G G G $

b

t

v

&

t v

$

$

G


31/169

9.4. PLANE QUARTICS 31

ThusB $

R $

b $ $form a polar triangle of

2

( . Since the associated conic is

irreducible no three points among theB $ t v R

s are linearly dependent. Thus2

(

is a Fermat cubic.

Lemma 9.4.2. Let(

Y

. Assume that

2

(

b cfor any

$

Y

.

Let

be the locus of points Y

such that the first polar of

( is a Fermat

cubic or belongs to the closure of its orbit. Then

is a plane quartic.

Proof. Let

V

Abe the Clebsch invariant vanishing on the locus of

Fermat cubics. It is a polynomial of degree 4 in coefficients of a cubic. If the

cubic is written in a Weierstrass form( b D E D

D

D

E

D

D

E

b c

, then

( b , for some nonzero constant

independent of

(.

Compose

with the polarization map

$

&

$ ( 0

2 4

( . We get a bihomogeneous map of degree

$ u

A. It

defines a degree 4 homogeneous map

ScV

(9.12)

This map is called the Clebsch quartic covariant. It assigns to a quartic form in

3 variables another quartic form in 3-variables. By construction, this map does

not depend on the choice of coordinates. Thus it is a covariant of quartics, i.e. a

GL

-equivariant map from

to some

. By definition, the locus of

Y

such that Sc

(

b cis the set of vectors

Y

such that

2

( b c,

i.e.,

2

( belongs to the closure of the Fermat locus.

Example 9.4.1. Assume that the equation of(

is given in the form

( b D

E

D

D

D

D

D

E

D

D

E

D

G

Then the explicit formula for the Clebsch covariant gives

Sc

( b

f

D

E

f

D

f

D

f

D

D

f

D

E

D

f

D

E

D

$

where

f

b

f

b

f

b

f

b

%

%

f

b

%

%

f

b

%

%

For a general(

the formula for

is too long.


32/169


Note that the map Sc defines a rational map

ScV

%

(9.13)

We call it the Scorza map in honor of Gaetano Scorza who studied the geometry of

this map. Note that the Scorza map is not defined on the closed subset of quartics

(

such that

2

(

belongs to the closure of the Fermat locus for any

Y

.

Proposition 9.4.3. The Scorza map is not defined on

( if and only if

( is

a Clebsch quartic admitting a reducible apolar conic.

We refer for a proof to [DK].

For any quartic curve%

satisfying the assumption of the previous proposition,

the curve Sc

% will be called the Scorza quartic associated to

%. If

%is a nonde-

generate Clebsch quartic, then, as we explained in above, the vertices of its polar

pentagon must belong to the Scorza quartic Sc

% . This gives

Proposition 9.4.4. Let ( be a nondegenerate Clebsch quartic. Then each polar

pentagon of(

is inscribed in the quartic curve

Sc

( .

Lemma 9.4.5. A quartic curve

which can be circusmscibed about a pen-

tagon defined by 5 lines

$ t

can be written in the form

b $

$

t

t

$ t

G

Proof. Consider the linear system of quartics passing through 10 vertices of a pen-

tagon. The expected dimension of this linear system is equal to 4. Suppose it is

larger than

. Since each side of the pentagon contains 4 vertices, requiring that a

quartic vanishes at some additional point on the side forces the quartic contain the

side. Since we have 5 sides, we will be able to find a quartic containing the union

of 5 lines, obviously a contradiction. Now consider the linear system of quartics

whose equation can be wriitten as in the assertion of the lemma. The equations have

5 parameters and it is easy to see that the polynomials$

$

$t

$ b $ G G G $ $are

linearly independent.

Definition 9.4.2. A plane quartic circumscribed about a pentagon is called a

Luroth quartic.

Thus we see that for any Clebsch quartic%

the Scorza quartic Sc

%

i s a Luroth

quartic. One can prove that any Luroth quartic is obtained in this way from a unique

Clebsch quartic (see [DK]). Since the locus of Clebsch quartics is a hypersurface


33/169


34/169


35/169


Proof. HereB

R $ B

Rdenote the class of

$

in the group Num

of divisor

classes on the surface

modulo numerical equivalence (or in

$

).Let

$

be the classes of fibres of the projections. For any

Y

the restriction

of the divisor class of

%

to each fibre

is equal to the restriction of a

divisor class

, where does not depend on . Thus

%

%

restricts to the trivial divisor class on each fibre

. This implies that

%

%

b

f

for some divisor class on

(see [Hartshorne], Ex. 12.4). Thus

we obtain the equality

B

R b

B

R

in Num

for some integers

$ . Intersecting with

we get5

b

.

Intersecting with

, we get5

b

. Intersecting with

B

Rand using the well-

known fact from topology thatB

R

b % , we get

B

R

B

R b

% .

Thus b 5

%

$

b 5

%

andB

R

B

R b

%

5

5

%

b

5

5

%

G

Proposition 9.4.8. Let % b

( be a general plane quartic. Then

is a finite

symmetric correspondence of degree

$

on

bSc

% without fixed points and

valency

.

Proof. The symmetry of

is obvious. We have a map from

to the closure

of the Fermat locus defined by 0

2

( . For any curve in

, except the

union of three lines, the set of points such that the first polar is a double line is

finite. It is equal to the set of double points of the Hessian curve and consists of 3

points for Fermat curves, one point for cuspidal cubics and 2 points for the unions

of a conic and a line. If(

is general enough the image of

in

does not intersectthe locus of the unions of three lines (which is of codimension 2). Thus we see

that each projection from

to

is a finite map of degree 3. Its branch points

correspond to the intersection of the image of

with the boundary of the orbit of

Fermat curves.

For any general point Y

the first polar2

% is a Fermat cubic. The divisor

consists of the three vertices of its unique polar triangle. For any

Y

the side

b

$ opposite to

is defined by2

2

% b 2

2

% b $

. It

is a common side of the polar triangles of2

% and

2

% . We have

b

, where

b T $

$

aand

b T $

$

a. This

gives

b

%

%

Y

E

G

Consider the map

V

Pic

given by

B

%

R

. Assume

is not

constant. If we replace in the previous formula

with

or

, we obtain that

b

b

b

E

%

G

Thus

is of degree6

on its image

and factors to a finite map to the normalization

of the image. Since a rational


36/169


37/169


Since

% b

%

f

implies that

f

(

f

, the assumption that

is not hyperelliptic implies that

is birational onto its image. It obviously blowsdown the diagonal to the zero point. For any

$

Y

, the divisor class

B %

R

is effective (use Riemann-Roch) and of degree %

. Let

be the divisor of

effective divisor classes in Pic

(the theta divisor) and

% its translate

in Jac

. We see that

. Let V

be the

switch of the factors. Then

b

b B % R

B % R

%

f

$

where

f

b

% . Since

b , the difference map

is injective on .

Thus

b

% b

%

f

G

Restricting to T a we see that the divisor classes and f

are equal. Hence

is a theta characteristic. By assumption,

E

b

E

%

b c

.

Note that a nonsingular plane quartic curve

is a nonhyperelliptic curve of

genus 3. This implies that, for any theta characteristic

on

,

E

is even (i.e.

is an even theta characteristic) and

E

b care equivalent properties. The

number of such theta characteristics is equal to

(see Part I).

Corollary 9.4.10. Let % be a general plane quartic curve and be the corre-

spondence on the Scorza curve

bSc

% . Then there exists a unique even theta

characteristic

such that b

.

Recall from Part I that an even theta characteristic

on a nonsingular plane

quartic b

( defines a net

of quadrics in

b , where

is the

divisor class of a line. It can be naturally identified with the plane

. The the

discriminant variety of singular quadrics is equal to the curve%

. The set of such

nets up to the natural action of GL

is a cover of degree

over the space of

nonsingular quartics. Let

ev denotes the normalization of

in

the field of rational functions on this cover. We have a finite morphism

ev

of degree 36 which is not ramified over the open subset of nonsingular quartics. Its

fibre over a nonsingular quartic

can be naturally identified with the set of even

theta characteristics on

.Example 9.4.2. (Bert van Geemen) Let

%

be a Clebsch quartic. Then Sc

%

is

a Luroth quartic, and by above it comes with a special even theta characteristic

. We know that

defines a representation of Sc

% as the determinant of

symmetric matrix with linear forms as its entries. It can be done explicitly.


38/169


If% b $

$

$

$

$

, then just take the

matrix

all of whose

entries are$

and add to this matrix the diagonal matrix with entries$

$ $

$ $

$ $

. Itsdeterminant is sum of all products of 4 of the 5 linear forms. It defines the Luroth

quartic.

.determinantal We omit the proof of the following theorem of G. Scorza whose

modern proof can be found in [DK]:

Theorem 9.4.11. Let Sc V

%

be the Scorza map. Then

the map% 0

Sc

% $ , where

b

, is a birational map from

to

ev. In particular, the degree of the Scorza map is equal to 36.

Remark 9.4.1. The Scorza theorem generalizes to genus 3 the fact that the map

from the space of plane cubics to itself defined by the Hessian is a birational

map to the cover

ev, formed by pairs

$ , where

is a non-trivial

2-torsion point (an even characteristic in this case). Note that the Hessian covari-

ant is defined similarly to the Clebsch invariant. We compose the polarization map

with the discriminant invariant

A

.

Under certain assumptions, which have not been yet verified, Scorza defines a

map from the space of canonical curves of genus

in

to the space of quartic

hypersurfaces in

(see [DK]).

9.4.3 Duals of homogeneous forms

We need to introduce here for the future use a classical construction which gener-

alizes the notion of a dual quadric to homogeneous forms of arbitrary even degree.Let

Y

and(

Y

Assume that

Cat

(

b c

. Then

there exists a unique form& Y

such that

2 4

( b

. This form is called

the anti-polar of

with respect to(

. This terminology extends immediately to

hypersurfaces defined by the forms.

To find&

we solve the linear equation Cat

(

B

&

R b B

R, where the brackets

mean that we choose a column of coordinates of&

and

with respect the bases in

and

used to compute the catalecticant matrix.

Let adjCat

( b

"

be the adjugate matrix of cofactors of the catalecticant

matrix, and

b )5

#

D

. Then the equation for the anti-polar (up to a constant

factor) is&

b )

)

b )

"

)

"

"

$(9.15)

where we choose the coordinates t b T t

in

.

The following lemma follows immediately from (9.15)


39/169


Lemma 9.4.12. Let

( b

"

"

"

Y

and

$

Y

. Then

(

b c 24

( b

t $

&

t

t b c

for some&

t

Y

$ b $

We say that&

is the anti-polar of a linear form$

if&

is the anti-polar form

of$

. A hyperplane

b

$ is called a bad hyperplane with respect to

(if

its anti-polar vanishes at

. It follows from the previous lemma that

is a bad

hyperplane if and only if

(

B $ R b c. Thus the hypersurface

( defines

the locus of bad hyperplanes.This is a hypersurface of degree

in

whose coefficients are homo-

geneous polynomials of degree

P

% . The form

( is a contravariant of

degree

and order

P

% on the space

.


. Then(

is a quadric and Cat

( is its discriminant.

The anti-polar is the dual quadric. Thus we have defined a generalization of the

dual of a homogeneous form of any even degree. If we consider the Veronese map

of degree

, then forms of degree

correspond to quadrics in the space

,

the dual form correspond to the dual quadric in this space.

9.4.4 Polar hexagons

A general quartic admits polar hexagons. Counting constants shows that the ex-

pected dimension of the variety

(

is equal to 3. Let us confirm it.

Proposition 9.4.13. Let% b

( be a general plane quartic curve. Then

(

is an irreducible variety of dimension 3.

Proof. Let

( $

b T

T B $

R $ G G G $ B $

R a $

Y

( $

V

Y

T B $

R $ G G G $ B $

R a G

Consider the projection to the second factor and look at its fibres. Fix one hexagon

T B $

R $ G G G $ B $

R a

containing a hyperplane

b

$

. Suppose we have anotherhexagonT B

R $ G G G $ B

R acontaining

. Then we can write

( b $

t

t $

t

b $

t

t

t

G


40/169


41/169


system vanish at the points

$ G G G $

. Counting constants we see that the di-

mension of the space of quartics passing through these points is equal to 8 (it couldbe larger only if 5 points are on a line). We have

2

4

( b 2

2 4

( b 2

$

b c $

2

4

( b 2

2 4

( b 2

$

b c G

This shows that linear system (9.16) is contained in the linear system of apolars

AP

( . This checks that

T

$ G G G $

a

is a polar hexagon.

The converse is also true. If we fix one line

b

$

in a polar hexagon, then

the conic

&

passing through the remaining 5 lines satisfies

2 4

( b

$

and hence is anti-polar conic of

. If we choose a second line

b

$

and

consider the conic

&

passing through

$

$

$

$

, then2 4

( b

$

and hence &

is an anti-polar of

contained in the pencil of conicsthrough

$

$

$

.

Note that something may go wrong in this construction. For example, the anti-

polar conics

&

$

&

may be reducible or may not intersect non-transversally,

or

may be on the conic

&

. But it is clear that this all could be avoided by

requiring that

is general enough.

Consider the set

(

ordof ordered polar hexagons. It is a Galois cover of

the space

(

with the Galois group

. The projection

$ G G G $

0

defines a map

(

ord

. It follows from the construction above, that

the hexagon can be reconstructed from a point

and a point

on the anti-polar

conic of

. Thus outside the quartic curve

(

the projection has fibres

isomorphic to open subsets of a conic. In other words,

(

ord has a birationalstructure of a conic bundle over

.

9.4.5 The variety of polar

-gons of a curve of degree .

Let

b

and(

Y

define a plane curve of degree

in

a . We

assume that AP

( b T c a

, i.e.

Cat

(

b c

. Consider the map

defined by&

0 24

(

. The expected dimension of the kernel is

.

We assume that it is the case by assuming further that the catalecticant matrix

Cat

(

is of maximal rank. Thus we have

2

( b

G

Let

b

. We consider the set

(

. Suppose

( b

t

$

t

G(9.17)


42/169


43/169


Next we want to find the image of the map (9.19). We follow Mukais idea from

section 9.3.2. Consider the following bilinear alternating map t v $ c 7

7

on the space

t v

$ b

(

Tt

Tv

% Tt

Tv

G

Note the polynomial in the brackets belongs to the space

and the polyno-

mial

(

Y

b

. SupposeB $

R $ G G G $ B $

Ris a polar polyhedron

of(

. Then

t v

$

$ $

b

(

$

$

$

for some constant

.

Lemma 9.4.15. For any 7

7

,

(

$

t

$

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Topics in Classical Algebraic Geometry