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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
"
$
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10 CHAPTER 9. APOLARITY
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(
.
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9.1. APOLAR SCHEMES 11
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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12 CHAPTER 9. APOLARITY
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
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9.1. APOLAR SCHEMES 13
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.
Example 9.1.3. Let b
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.
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14 CHAPTER 9. APOLARITY
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
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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)
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16 CHAPTER 9. APOLARITY
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
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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
(
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18 CHAPTER 9. APOLARITY
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
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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
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20 CHAPTER 9. APOLARITY
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
(
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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 $
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22 CHAPTER 9. APOLARITY
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
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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.
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24 CHAPTER 9. APOLARITY
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
$
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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.
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26 CHAPTER 9. APOLARITY
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,
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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
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28 CHAPTER 9. APOLARITY
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
.
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30 CHAPTER 9. APOLARITY
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
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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.
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32 CHAPTER 9. APOLARITY
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
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9.4. PLANE QUARTICS 35
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
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9.4. PLANE QUARTICS 37
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.
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38 CHAPTER 9. APOLARITY
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)
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9.4. PLANE QUARTICS 39
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
.
Example 9.4.3. Let b
. 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
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9.4. PLANE QUARTICS 41
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)
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9.4. PLANE QUARTICS 43
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
$