Determinantal and Pfaffian Hypersurfaces

Introduction
The study of cubic surfaces in P
3
is a classical subject. According to
Meyer ([Me928]) it originated from the works of Plücker ([Pl829]) on inter-
section of quadrics and cubics and Magnus ([Ma833]) on maps of a plane
by a linear system of cubics. However, it is a common thought that the
theory of cubics surfaces started from Cayley’s and Salomon’s discovery of
twenty-seven lines on a nonsingular cubic surface ([Ca849]). Salmon’s proof
was based on his computation of the degree of the dual surface ([Sa847]),
while Cayley’s proof used the count of tritangent planes through a line. In
1851 Sylvester claimed without a proof that a general cubic surface can be
written uniquely as a sum of five cubes of linear forms ([Sy851]). This fact
was proved ten years later by Clebsch ([Cl861]); we briefly mention it in sec-
tion 2.5. In 1854 Schläfli discovered thirty-six double-sixers on a nonsingular
cubic surface. In 1866 Clebsch proved that a general cubic surface can be
obtained as the image of a birational map from the projective plane given by
cubics through six points ([Cl866]). Using this he showed that the Schläfli’s
notationa
i
;b
j
;c
ij
for twenty-seven lines correspond to the images of the ex-
ceptional curves, conics through five points and lines through two points.
This important result was independently proved by Cremona in his memoir
([Cr868]) of 1868. A modern treatment of the cubic surface — mainly on
the complex field — as image of a birational map can be found especially in
[Ha977], even though in terms of some advanced pieces of theory. We refer
to [Ge989] for a quite easy construction of the cubic surface.
However, many mathematicians contributed to the understanding of smooth
cubic surfaces: one can find very much in [He911], [Re988] about the geome-
try of the twenty-seven lines and in [HC952] about the geometry of Schläfli’s
double-sixers.
The representation of codimension one curves and surfaces of small de-
gree as linear determinants is a classical subject too. The case of cubic
surfaces was already known in the middle of nineteenth century ([Gr855]);
other examples of curves and surfaces are treated in [Sc881]. The general
homogeneous forms which can be expressed as linear determinants are deter-
mined in [Di921], where Dickson showed that every curve has a determinan-
tal representation. Determinantal representations in general were studied by
manyauthors, forinstanceBeauville([Be000]), CookandThomas([CT979]),
11
Introduction 12
Room ([Ro938]). We refer especially to the first one for a modern develop-
ment of determinantal representations of hypersurfaces.
The case of curves has been recently studied for example by Vinnikov.
Nonequivalent linear determinantal representations ofC are in bijection with
certain line bundles on the curveC (already known in [CT979]) and they can
be parametrized by the non exceptional points of the Jacobian variety
fline bundlesL= degL =
1
2
d(d 3);h
0
(C;L) = 0g
[Vi989]. To any compact Riemann surface X one can associate the pair
(JX; ) , the Jacobian and the Riemann theta function. The geometry of
the pair is strongly related to the geometry of X. This gives an idea that
higher rank vector bundles define a non-abelian analogue of the Jacobian
called moduli space firstly due to the mathematicians of the Tata Institute
[Ne978]. MuchlaterphysicistsinConformalFieldTheoryintroducedpairsof
moduli spaces and determinant line bundles on these moduli spaces [Th992].
This has made a clear analogy with the Jacobian pair.
The case of determinantal representations of cubic surfaces was developed by
Cremona [Cr868]. Such determinantal representations have been studied for
example by [BL998], [Ge989]. Recently, Brundu and Logar have considered
singular cubic surfaces and have showed that all, except those containing
one line only, have determinantal representations (see also subsection 2.5.1).
In [Be000] we find the following proposition, reported by us as proposition
3.3.2:
Proposition 1. LetC be a projectively normal curve on S [a smooth cubic
surface in P
3
], of degree
1
2
d(d 1) and genus
1
6
(d 2)(d 3)(2d + 1). The
line bundleO
S
(C) admits a minimal resolution
0
//O
d
P
3
( 1)
M
//O
d
P
3
//O
S
(C)
// 0
with det(M) =F.
Conversely, letM be a linear determinantal representation forS; the cok-
ernel of M :O
d
P
3
( 1)
//O
d
P
3
is isomorphic toO
S
(C), whereC is a smooth
projectively normal curve on S with the above degree and genus.
Also Buckley and Košir, recently, have worked on this one-to-one corre-
spondence. In [BK007] we can found
Theorem 2. There is a one-to-one correspondence between classes of deter-
minantal representations of S and sixers of lines on S. In particular, there
are 72 different classes.
reported by us as theorem 2.3.1. In particular, Beauville has proved that
determinantal representations of a smooth cubic surface are in one-to-one
correspondence to particular line bundles associated to twisted cubic curves
Introduction 13
on the surface; in this way there is a link between determinantal representa-
tions and ACM sheaves. This gives also a reason for our choice, because only
curves and cubic surfaces admit generically a determinantal representation,
as a consequence of Lefschetz’s and Severi’s theorems (see page 64).
A motivation to study determinantal representations comes from possi-
ble application to multiparameter spectral theory ([Ko003]). Classification
of (selfadjoint) determinantal representations of algebraic hypersurfaces is
equivalent to simultaneous classification of t-uples of (selfadjoint) matrices.
Classically, the existence of determinantal representations allow to deduce
the rationality of some moduli spaces. Vinnikov ([Vi989], [Vi993]) stressed
another very important motivation to study self-adjoint and definite deter-
minantal representations. They appear as determinantal representations of
discriminant varieties in the theory of commuting nonselfadjoint operators
in an Hilbert space ([LKMV]).
Study of pfaffian representations is strongly related to and motivated by
determinantal representations: we can consider the case of determinantal
representations as a particular case of pfaffian ones, because from a matrix
M with det(M) =F we can construct
 0 M
 M
t
0
 ,
which is a pfaffian representation of the same variety F = 0.
In linear algebra, pfaffians have been studied for many purposes; one can see
[BE977] for a modern treatment, which also works with projective modules.
Moredetailsonformulasinvolvingpfaffianscanbefoundin[Bo970], [He969],
[MO003], [FP998].
Study of pfaffian representations is not very developed, even though recently
it has been strongly reconsidered. In 1996 Adler proved ([AR996]) that a
general cubic threefold can be written as a linear pfaffian. We refer mainly
to [Be000], who proved that plane curves of any degree, surfaces of degree
less than sixteen and threefolds of degree less than six can be generically
defined by a linear pfaffian.
ThecaseofcurveshavebeenapproachedalsobyBuckleyandKosir([BK008],
[Bu009]); given a curveC, they find all the linear pfaffian representations of
C and relate them to the moduli space M
C
(2;K
C
) of semistable rank two
vector bundles onC with canonical determinant. They also consider not
necessarily linear pfaffian representations. The case of surfaces has not been
studied in details; Beauville is the first to link linear pfaffian representations
to AG sub-scheme, providing, for a surface of degree d,
Proposition 3. Let S be a smooth surface of degree d in P
3
. The following
conditions are equivalent:
1. S can be defined by an equation Pf(M) = 0, where M is a skew-
symmetric linear (2d 2d)-matrix;
Introduction 14
2. S contains an AG subscheme Z of index 2d 5, not contained in any
surface of degree d 2.
More precisely, under the hypotheses of 2., the rank 2 vector bundle E asso-
ciated to Z admits a minimal resolution
0
//O
2d
P
n
( 1)
M
//O
2d
P
n
//E
// 0;
Moreover, the degree of Z is
1
6
d(d 1)(2d 1).
which is reported as proposition 4.2.1. The whole study of not necessarily
linear pfaffian representations is strongly linked to rank 2 ACM vector bun-
dles and their free resolutions; for a cubic surface, they have been studied in
details by Faenzi ([Fa008]).
The aim of the work was to explain geometrically the meaning of the ex-
istence of determinantal or pfaffian representations of a smooth, irreducible,
cubic surface inP
3
on an algebrically closed field of characteristic zero; more
generally, to analyze Beauville’s approach to hypersurfaces in [Be000]. The
cubic surface is indeed the first significative case, because curves have been
already treated by Buckley and Košir.
A brief outline of the work is the following; throughout the work, we deal
mainly with a smooth, irreducible, cubic surface in P
3
. In chapter one, fol-
lowing [Ge989], we start from the blow-up of P
2
along six distinct points
in general position and we show that it is isomorphic to a cubic, smooth,
irreducible surface in P
3
. We show also that, via Hilbert-Burch theorem, it
is possible to give a linear determinantal representation of the surface.
In chapter two, following mainly [BK007] and [Ge989], we prove the one-
to-one correspondence between classes of determinantal representations of a
cubic surface under the action of GL
3
(K) GL
3
(K) and sets of six skew lines
on the surface (theorem 2.3.1); then we prove the one-to-one correspondence
between them and linear systems of twisted cubic curves on the surface,
according to the approach of Buckley and Kosir . Then we consider some
concrete problems from a computational point of view e we apply theory to
Fermat cubic and Clebsch cubic.
In chapter three, following [Be000] and [Do010], we introduce ACM sheaves
and we give a proof of the one-to-one correspondence between classes of de-
terminantal representations of a surface of degree d and linear systems of
curves of degree
1
2
d(d 1) and genus
1
6
(d 2)(d 3)(2d + 1) on the surface
(proposition 3.3.2). Then we specialize to cubic surfaces and we find again
the one-to-one correspondence between classes of determinantal representa-
tions and linear systems of twisted cubic curves on the surface.
In chapter four, following [Be000], we introduce arithmetically Gorenstein
sub-scheme and we prove for a surface of degree d inP
3
the equivalence be-
tween the existence of a linear pfaffian representation and the existence of
Introduction 15
an AG sub-scheme of index 2d 5, not contained in any surface of degree
d 2 (see proposition 4.2.1).
In chapter five, we prove the Buchsbaum-Eisenbud Theorem
Theorem 4.
1. Let n 3 be an odd integer, and let F be a free R-module of rank
n. Let f :F
// (F )
 be an alternating map of rank n 1 whose
image is contained in J(F )
 . If Pf
n 1
(f) has grade 3, then Pf
n 1
(f)
is Gorenstein, and the minimal number of generators of Pf
n 1
(f) isn.
2. Every Gorenstein ideal of R of grade 3 arises as in 1.
(reported as theorem 5.2.1) which is an analogous of Hilbert-Burch’s The-
orem. Applying this theorem, we give an explicit algorithm to get a linear
pfaffian representation for every cubic surface containing five points in gen-
eral position. This is obtained starting from the ideal of the five points:
one proves that this ideal is Gorenstein in the sense of definition 5.1.2 and,
making use of the proof of Buchsbaum-Eisenbud Theorem, one takes a free
resolution of the ideal and builds a (5 5) skew-symmetric matrixT. “Aug-
menting” T by a column (and a row) it is possible to obtain the desired
equation. We finally apply this algorithm to Fermat cubic and Clebsch cu-
bic.
Chapter 1
The cubic surface as a blow-up
In this chapter we see that blowing-up the planeP
2
in six points we get
a smooth, irreducible, cubic surface in P
3
and then, making use of Hilbert-
Burch Theorem, we are able to find a determinantal representation of the
surface. We refer mainly to [Ge989].
From now on, let K be an algebraic closed field of characteristic 0; we
will useP
n
to indicate, as usual, the projective spaceP
n
K
.
1.1 Six points in P
2
Let us consider six distinct points P
1
;P
2
;:::;P
6
in P
2
such that they
do not lie on a conic and no three of them lie on a line. We set X =
fP
1
;P
2
;:::;P
6
g.
1.1.1 Some properties of I(X)
Our first aim is to find some useful properties of the homogeneous ideal of
thesixpoints, I(X) K[x
0
;x
1
;x
2
], generatedbythosepolynomialsvanishing
simultaneously at P
1
;P
2
;:::;P
6
. We consider a point P
i
and its ideal p
i
=
I(fP
i
g), generatedbytwolinearformsL
i1
;L
i2
; itiseasytoprovethat I(X) =
p
1
T
p
2
T
 T
p
6
. We have the following
Properties 1.1.1.
1. ht(I(X)) = 2, where ht(I) means the height;
2. A =K[x
0
;x
1
;x
2
]=I(X) is a Cohen-Macaulay ring;
3. hd I(X) = 1, where with hd we mean the homological dimension in
K[x
0
;x
1
;x
2
].
Proof.
16
1.1 Six points in P
2
17
1. ht(I)isdefinedastheminimumheightoftheprimeidealscontainingI.
So, considering the inclusion I(X) p
i
, we have ht(I(X)) ht(p
i
) = 2.
The equality follows from the prime avoidance lemma: if a prime p
contains I(X), then we have
p p
1
\
p
2
\
 \
p
6
=) p p
i
for some i
and so we have ht(I(X)) = 2;
2. the ringA is Cohen-Macaulay if and only if depth(A) = dim
K
(A), the
Krull dimension ofA. Since we have proved that ht(I(X)) = 2, we have
dim
K
(A) = dim
K
(K[x
0
;x
1
;x
2
]) ht(I(X)) = 3 2 = 1
and it is sufficient to prove that depth(A) = 1, that is there exists a
non zero-divisor in A.
Let L be a linear form not vanishing in any of the P
i
, that is, a line
through no one of the six points. So L = 2 p
i
for any i and, from the
relation I(X) = p
1
T
p
2
T
 T
p
6
, we have that
L F2 I(X) =) L F2
\
i
p
i
=) F2
\
i
p
i
,
hence L is not a zero-divisor in A.
3. it is a general fact that holds for finite sets of points, it can be found
for example in [Ei995, 20.4].
Let L be a (n m)-matrix of (homogeneous) forms; we will use the
symbolI
p
(L) to indicate the ideal generated by the (p p)-minors of matrix
L.
The following theorem is stated and proved for example in [Ei995, 20.4].
Hilbert-Burch Theorem. If a complex
0
//F
L
//R
n
G
//I
// 0 (1.1)
is exact, then F
 = R
n 1
and there exists a nonzerodivisor a such that I =
aI
n 1
(L). In fact the ith entry of the matrix G is ( 1)
i
a times the minor
obtained from L by leaving out the ith row. The ideal I
n 1
(L) has depth
exactly two.
Conversely, given any (n 1) n-matrix L such that depth(I
n 1
(L)) 2,
and a nonzerodivisora, the mapG obtained as above makes the sequence 1.1
into a free resolution of I =aI
n 1
(L).
1.1 Six points in P
2
18
In our context, it will be useful the following slightly different formula-
tion, stated for example in [KR005, Tutorial 72].
Hilbert-Burch Theorem (II). Let R =K[x
0
;x
1
;:::;x
m
] and let I be an
homogeneous ideal inR with ht(I) = 2 and hd
R
I = 1. Let the minimal num-
ber of generators  (I) = n, then I admits a graded minimal free resolution
of the form
0
//R
n 1
L
//R
n
G
//I
// 0
and is generated by the maximal minors (n 1) (n 1) of an n (n 1)-
matrix L of homogeneous forms. The ith entry of the (1 n)-matrix G is
( 1)
i
times the minor obtained from L by removing the ith row. Moreover,
any minimal system of generators of I can be represented in this way.
Let (I(X)) =n and let us set a minimal system of homogeneous genera-
tors (g
0
; g
1
;:::; ( 1)
n 1
g
n 1
) of I(X). We can apply the previous theorem
to I(X), because all the hypotheses are satisfied due to the properties 1.1.1.
We get the free resolution
0
//R
n 1
L
//R
n
G
// I(X)
// 0,
(1.2)
where G =
 g
0
g
1
::: g
n 1
 .
Let us say more about L and G.
Property 1.1.2. There are no non-zero entries of degree zero in L, that is
the entries of L do not belong to Knf0g.
Proof. The proof is by reductio ad absurdum. Let us suppose thatL = (L
ij
)
has a non-zero entry of degree zero, namely L
00
. The free resolution 1.2
implies that
n 1
X
i=0
L
i0
g
i
= 0,
but then
g
0
=
n 1
X
i=0
L
i0
L
00
g
i
and so the g
i
are not a minimal system of generators of I(X), contradiction.
Observation 1.1.3. Let us remark that the previous property still holds for
every ideal satisfying the hypoteses of Hilbert-Burch Theorem.
Hilbert-Burch Theorem has also the beautiful “consequence that an ideal of
a set of points with many generators must have all generators of rather high
degree; this was the application that Burch originally had in mind” ([Ei995,
20.4]). We will see what it means in property 1.1.4: for an ideal I satisfying
the hypotheses of Hulbert-Burch Theorem, having a number of generators
 (I) n makes these generators to be of degree at least n 1.
1.1 Six points in P
2
19
Property 1.1.4. I(X) is minimally generated by four (linearly independent)
cubic forms.
Proof. Recall that, if we consider the graded ringK[x
0
;x
1
;x
2
] =
L
i
R
i
and
the homogeneous ideal I(X), we can write the quotient ring A as direct sum
of its homogeneous componentsA =
L
i
A
i
. Let us consider now the Hilbert
function of A
1
H(A;t) = H(X;t) = dimA
t
= dimR
t
 dim I(X)
t
=
8
<
:
1 if t = 0
3 if t = 1
6 if t 2
,
as we obtain from the hypothesis onX. In fact we have trivially that
dimA
0
= dimR
0
 dim I(X)
0
= 1 0 = 1,
moreover dim I(X)
1
= dim I(X)
2
= 0 and so
dimA
1
= dimR
1
=
 3 + 1 1
1
 = 3,
dimA
2
= dimR
2
=
 3 + 2 1
2
 = 6.
The set X contains six points, so H(A;t) = 6 also for t > 2. In particular,
we have H(A; 3) = 6 and so dim(I(X))
3
= 4.
Let us set J = (I(X))
3
. There exist four cubic forms vanishing on X such
that they form a basis for J as vector space. Since (I(X))
1
= (I(X))
2
=f0g,
it is clear that every minimal system of generators of I(X) must contain a
basis of J and so  (I(X)) 4.
If  (I(X)) = n 5, from Hilbert-Burch Theorem and by property 1.1.2 we
would have that the degree of all the generators is at least 4, but then it
would be impossible for such a system of generators to contain a basis of J.
Therefore we can conclude that  (I(X)) = 4 and that I(X) is generated by
four linearly independent cubic forms.
Observation 1.1.5. The hypothesis we made about the position of the points
is essential here. For example, let us consider six points on a conic. The
associated ideal is generated by the conic itself and a cubic passing through
them. Of course the ideal has a system of generators composed by four
cubics, but it is not minimal.
Let us see what we get from Hilbert-Burch Theorem applied to our ideal
I(X). Chosen a minimal system of generatorsfF
0
; F
1
;F
2
; F
3
g, we get the
free resolution
0
//R
3
L
//R
4
G
// I(X)
// 0,
(1.3)
1
We will say, as well, the Hilbert function of X.