Heisenberg invariant quartics and SUC for a curve of genus four

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Niveau: Supérieur, Doctorat, Bac+8
Heisenberg invariant quartics and SUC(2) for a curve of genus four William Oxbury and Christian Pauly The projective moduli variety SUC(2) of semistable rank 2 vector bundles with trivial determinant on a smooth projective curve C comes with a natural morphism ? to the linear series |2?| where ? is the theta divisor on the Jacobian of C. Well-known results of Narasimhan and Ramanan say that ? is an isomorphism to P3 if C has genus 2 [16], and when C is nonhyperelliptic of genus 3 it is an isomorphism to a special Heisenberg-invariant quartic QC ? P7 [18]. The present paper is an attempt to extend these results to higher genus. In the nonhyperelliptic genus 3 case the so-called Coble quartic QC ? |2?| = P7 is characterised by either of two properties: (i) QC is the unique Heisenberg-invariant quartic containing the Kummer variety, i.e. the image of Kum : JC ? |2?|, x 7? ?x + ??x, in its singular locus; and (ii) QC is precisely the set of 2?-divisors containing some translate of the curve W1 ? J1C . We shall examine, for a curve of genus 4, the analogue of each of these properties, and our first main result, analogous to (i), is the following.

  • theta characteristic

  • invariant quartic

  • jg?1 ?

  • qc ?

  • any great

  • wg?d ? supp

  • embedded prym- kummer variety

  • without vanishing


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Heisenberg invariant quartics andSUC(2) for
a curve of genus four

William Oxbury and Christian Pauly

The projective moduli varietySUC(2) of semistable rank 2 vector bundles
with trivial determinant on a smooth projective curveCcomes with a natural
morphismφto the linear series|2Θ|where Θ is the theta divisor on the
Jacobian ofCresults of Narasimhan and Ramanan say that. Well-knownφ
3
is an isomorphism toPifChas genus 2 [16], and whenCis nonhyperelliptic
of genus 3 it is an isomorphism to a special Heisenberg-invariant quartic
7
QC⊂P[18]. Thepresent paper is an attempt to extend these results to
higher genus.
In the nonhyperelliptic genus 3 case the so-calledCoble quarticQC⊂
7
|2Θ|=Pis characterised by either of two properties:
(i)QCis the unique Heisenberg-invariant quartic containing the Kummer
variety, i.e.the image of Kum :JC→ |2Θ|,x7→Θx+ Θ−x, in its
singular locus; and
(ii)QCis precisely the set of 2Θ-divisors containing some translate of the
1
curveW1⊂J.
C
We shall examine, for a curve of genus 4, the analogue of each of these
properties, and our first main result, analogous to (i), is the following.Recall
that a curve of genus 4 has no vanishing theta-nulls if and only if its canonical
3
model inPlies on a nonsingular quadric surface; equivalently if it carries
two distinct trigonal pencils.

0.1 Theorem.IfCis a curve of genus 4 without vanishing theta-nulls then
there exists a unique (irreducible) Heisenberg-invariant quarticQC⊂ |2Θ|=
15
Pcontainingφ(SUC(2))in its singular locus.

We prove this in sections 3 and 4 (see corollary 4.2).The main work
involved is first to show cubic normality forφ(SUCWe(2)) (theorem 4.1).

1

then use the Verlinde formula to deduce that its ideal contains exactly sixteen
independent cubics; by symmetry considerations these cubics are the partial
derivatives, with respect to the homogeneous coordinates, of a single quartic
QC. Theargument here is identical to that of Coble [6] for the genus 3 case.
We conjecture thatφ(SUC(2)) = SingQC, or equivalently that the ideal
ofφ(SUC(2)) is generated by cubics.We cannot prove this, but in the
rest of the paper we examine the relationship of this problem with
property (ii) above.For any curve one may construct a sequence of irreducible,
Heisenberg-invariant subvarieties, for 1≤d≤g−1,

d−1
Gd={D|x+Wg−d⊂suppDfor somex∈J} ⊂ |2Θ|.
C
In particular,G1is the Kummer variety, whileGg−1is a hypersurface
containingφ(SUC(2)) and which coincides with the Coble quartic in the case
g= 3 (and with the Kummer quartic surface in the caseg= 2).When
g= 4, however,G3turns out to be distinct fromQC—quite contrary to our
original expectation.
7 15
We see this by restricting to the eigen-Ps of the action on|2Θ|=Pof
the groupJCFor any nonzero element[2] of 2-torsion points.η∈JC[2]
e
we have an associated double coverπ:C→Cwith respect to which

P, where (P ,Ξ) is the prin
ker Nm=Pη∪η ηcipally polarised Prym
vari7
ety. Thefixed-point set of theη-action on|2Θ|is a pair ofPs either of
which can be naturally identified with|2Ξ|; this fixed-point set therefore

contains the Kummer image ofPη∪P, and this is precisely the intersection
η
withφ(SUC(2)).
Beauville and Debarre [3] have shown that a|2Ξ|-embedded
PrymKummer variety admits a 4-parameter family of quadrisecant planes
analogous to the trisecant lines of a Jacobian Kummer.We prove:

0.2 Theorem.LetCbe a curve of genus 4 without vanishing theta-nulls;
and for any nonzeroη∈JC[2]identify|2Ξ|֒→ |2Θ|as the component of the
fixed-point set ofηcontaining the Kummer image ofPη. Then:

15
1.QC⊂Prestricts on|2Ξ|to the Coble quartic ofKum(Pη).
15
2.G3⊂Prestricts on|2Ξ|to the hypersurface ruled by the quadrisecant
planes ofKum(Pη); and this is distinct from the Coble quartic.

Part 1 is proved in section 4.Note thatPηis necessarily the Jacobian
JXof some curveXof genus 3, which can be constructed explicitly (given a

2

choice of trigonal pencil onC) via the Recillas correspondence (see section
2). Insection 5 we obtain necessary and sufficient conditions for a secant line
of Kum(JX) to lie onφ(SUX(2)); and in section 6 this is used to show that
a generic quadrisecant plane of the family does not lie onφ(SUX(2))—hence
the final remark in part 2 of the theorem.
We introduce the filtration of|2Θ|by the subvarietiesGdin section 7.
d−1
These varieties are ruled, asx∈Jvaries, by the subseriesN(x)⊂ |2Θ|
C
of divisors containingx+Wg−d. Moreover,for any theta characteristicκon
the curve the rulings ofGdandGg+1−dare polar in the sense that

−1
N(x)⊥N(κx)

with respect to the bilinear form on|2Θ|induced byκ(symmetric or
skewsymmetric accordingly asκOne would like to know thatis even or odd).
−1
the subspacesN(x) andN(κxHowever,) have complementary dimension.
this depends on the vanishing, which forg >4 we are unable to prove, of
d
certain cohomology groups on the symmetric productsS C. Thisproblem
and the associated computations are discussed in the appendix.
−1
For genus 4, though, we show thatN(x) andN(κx) are indeed
orthogonally complementary, and we arrive at a configuration:

15
Kum(J)⊂G2⊂φ(SUC(2))⊂G3⊂P

whereG2is a divisor inφ(SUC(2)) ruled by 4-planes, andG3is ruled by
their polar 10-planes (with respect to any theta characteristic).G2contains
the trisecants of the Jacobian Kummer variety.(This remains true for higher
genus, in fact.G2is the ‘g-plane ruling’ of [20] section 1; the trisecants are
analysed in [20] section 2.)The ruling ofG3, on the other hand, cuts out in
|2Ξ|, for eachη∈JC[2], precisely the Beauville–Debarre quadrisecant planes.
(Again it is true for any genus thatG3contains the quadrisecant planes.Note
that in general the smallestGdcontainingφ(SUC(2)) isG[g/2]+1.)
Acknowledgments.The authors are grateful to Arnaud Beauville and Bert
van Geemen for some helpful discussions; and to Miles Reid and Warwick
MRC for their hospitality during the early part of 1996, when much of this
work was carried out.The second author thanks the University of Durham
for its hospitality during 1996, and European networks Europroj and AGE
for partial financial support.

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