A remark on the Schottky locus in genus 4

6 Pages·0.08 MB·English

by  J. Harris

A remark on the Schottky locus in genus 4
Joe Harris and Klaus Hulek
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0 0 Introduction
2

n
By Torelli’s theorem the map t : M → A which associates to each curve
a g g g
J C its Jacobian variety J(C), together with the principal polarization given

2 by the theta divisor, is injective. We denote its image by J . In genus 4
g
Mumford[Mu]has determined the class of theclosure J′ of J in the partial
] 4 4
G compactification A′ which is given by adding the rank 1 degenerations. In
4
A thisbriefnoteweshalldeterminetheclass oftheSchottky locusintheIgusa
. and the Voronoi compactifications of A and we shall briefly comment on
h 4
t the degree 8 modular form which vanishes on the Jacobi locus J4.
a
m
1 The main result
[

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v We denote by AIgu and AVor the Igusa, resp. the second Voronoi compac-
g g
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tification of A . These are toroidal compactificiations given by choosing
0 g
0 either the second Voronoi decomposition (which gives the second Voronoi
1 compactification) or the central cone decomposition, which for genus g = 4
0
coincides with the perfect cone decomposition (and which leads to the Igusa
3
0 compactification). If g = 4, then AIgu is singular and AVor is smooth in
4 4
/ the stack sense. By this we mean the following: the varieties AVor and
h 4
t AIgu are both singular, but AVor only has finite quotient singularities and
a 4 4
m addingafulllevel-n structureof level n ≥ 3makes AVor(n)smooth, whereas
4
Igu
: A (n) will always be singular. These singularities come from the choice
v 4
Igu
i of the fan. As a stack A has exactly one singularity (and by abuse of
X 4
Igu
notation we will refer to this as thesingularity of A . Thereis a morphism
r 4
a π : AVor → AIgu which is a blow-up with centre at the singular pointof AIgu
4 4 4
(see [HS, chapter I]). (Note that this is special for genus 4.) We denote
by D′, resp. DIgu and DVor the boundary component given by the rank
4 4 4
1 degenerations, resp. its closure in AIgu and AVor and we denote by E
4 4
the (reduced) irreducible divisor in AVor which is exceptional with respect
4
to π : AVor → AIgu. Finally, let L be the (Q)-line bundle of modular
4 4
forms of weight 1 on the Satake compactification ASat. Since both AIgu
4 4
and AVor map to the Satake compactification we can also consider L as a
4
line bundle on these varieties, which, by abuse of notation, we will again
1
denote by L. We denote by A′ Mumford’s partial compactification which
4
is given by adding the rank 1 degenerations of principally polarized abelian
varieties. Note that AIgu\A′ has codimension 2 whereas AVor\A′ contains
4 4 4 4
the exceptional divisor E. We finally recall from [HS, Prop. I.1 and Prop.
I.6] that the inclusion i: A′ → AIgu induces an isomorphism
4 4
i∗ :Pic AIgu⊗Q = QL⊕QDIgu ∼= Pic A′ ⊗Q
4 4 4
and that
Pic AVor⊗Q = QL⊕QDVor⊕QE.
4 4
Moreover by [HS, Prop I.3]
π∗DIgu = DVor+4E.
4 4
The Torelli map t : M → A extends by [N] to a morphism t¯ : M →
g g g g g
AVor on the Deligne-Mumford compactification of M . We denote its image
g g
by JVor and we set JIgu = π JVor . This coincides with the closure of J′ in
4 4 4 4
AIgu. (cid:0) (cid:1)
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Theorem 1.1 The varieties AVor and AIgu are Q-factorial varieties. For
4 4
the Schottky locus the following holds:
(i) The Schottky locus JVor does not meet the exceptional divisor E in
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AVor and hence JIgu does not pass through the singular point of AIgu.
4 4 4
(ii) The divisor JIgu is Q-Cartier in AIgu and its class equals
4 4
Igu Igu
[J ] = 8L−D .
4 4
(iii) The divisor JVor is Q-Cartier in AVor and its class equals
4 4
[JVor]= 8L−DVor−4E.
4 4
Proof. We shall first prove that AVor and AIgu are Q-factorial varieties.
4 4
This is clear for AVor since it is the quotient of the smooth variety AVor(n),
4 4
for n ≥ 3 by a finite group, namely PSL(8,n). In particular also A′ is Q-
4
factorial and by Mumford’s result we know that Pic A′ ⊗Q is generated by
4
L and D′. Both these divisors extend to Q-Cartier divisors on AIgu where
4 4
they generate the group Pic AIgu⊗Q.
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Next we shall prove that JVor and the exceptional locus E have empty
4
intersection. Assume that this is not the case. Since E is a Q-Cartier
divisor on AVor this would imply that (t¯ )−1(E) has codimension 1. Recall
4 4
that the points on E all correspond to degenerations of abelian varieties
with trivial abelian part. This would imply (cf.[A]) that (t¯ )−1(E) is a Q-
4
divisorwhosegeneralpointforeachcomponentcorrespondstoastablecurve
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whose normalization has only rational components. But this is impossible
since M maps to A and the boundaryM \M consists of 3 components
4 4 4 4
M4−i,i;i = 0,1,2 whose general points consist of a curve of genus 3 with
one node, a curve of genus 3 and an elliptic curve intersecting in one point,
or two genus 2 curves intersecting in one point respectively. These are 8-
dimensional families, but their general point is mapped to D′ in the first
4
case, resp. theinterior A intheothertwocases. Inparticular,theJacobian
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of the general pointof such a divisor has an abelian part of dimension either
3 or 4.
It was shown by Mumford [Mu, p.369] that
′ ′
[J ] = 8L−D
4 4
and this, together with the fact that the inclusion i: A′ → AIgu induces an
4 4
isomorphismofthePicardgroups,proves(ii). Sinceπ∗DIgu = DVor+4E the
4 4
classofthetotaltransformofJIgu inAVor equalsπ∗(JIgu) = 8L−DVor−4E.
4 4 4 4
Igu Igu
But since J does not pass through the singular point of A this also
4 4
proves (iii). 2
2 Modular forms
There exists a weight 8 modular form F which vanishes precisely along
J . There are two ways to write this down explicitly. The first goes back
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to Igusa. Let S be an even, integer, positive definite quadratic form with
determinant 1 of rank m. Such a form defines a genus g modular form of
weight m/2 given by the theta series
ϑ(g)(τ) = exp[πiTr(tGSGτ)].
s
G∈MaXt(m,g;Z)
There is only one even, integer, positive definite quadratic form S(8) with
determinant 1 of rank 8 and there are two such formsS(8)⊕S(8) and S(16)
of rank 16. Igusa [I], cf. also [W, p.20] has shown that
(4) (4)
F(τ) = ϑ (τ)−ϑ (τ)
S(8)⊕S(8) S(16)
is a non-zero cusp form of weight 8 which vanishes on J .
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The second method goes back to van Geemen and van der Geer. For ε,ε′ ∈
{0,1}g−1 define the following theta functions
ε ε ε ε′
θ[εε′](τ,z) = exp[πit((m+ )τ(m+ )+2t(m+ )(z+ ))],
2 2 2 2
m∈XZg−1
resp. the theta constants
ε ε
θ[ε′] = θ[ε′](τ,0).
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Such a theta function is called even (resp. odd) if tεε′ ≡ 0 mod 2 (resp.
tεε′ ≡ 1 mod 2). The odd theta constants vanish identically and there are
2g−2(2g−1+1) even theta constants. They define a morphism
Sq: Ag−1(2,4) −→ PN, N = 2g−2(2g−1 +1)−1
[τ] 7−→ (...,θ2[εε′](τ,0),...).
Van Geemen [vG1], resp. van Geemen andvan derGeer [vGvG]have shown
how elements in the ideal of Sq(Ag−1(2,4)) give rise to modular forms van-
ishing on the Schottky locus J in A . This can be described as follows.
g g
Let C[X0,...,XN] be the coordinate ring of PN, and let Ig−1 be the homo-
geneous ideal of Sq(Ag−1(2,4)). Then for every element G(X0,...,XN) of
Ig−1 we define a form on Hg as follows
ε 0 ε 0
σ(G)(τ) = G(...,θ (τ,0)θ (τ,0),...); τ ∈H .
(cid:20)ε′ 0(cid:21) (cid:20)ε′ 1(cid:21) g
This will then be a modular form vanishing on J .
g
In the case of genus g = 4 we, therefore, have to start with a theta
relation for g = 3. According to [vGvG, p. 623] one has the following
relation
0 0 0 0 0 0 0 0 0 0 0 0
θ θ θ θ
(cid:20)0 0 0(cid:21) (cid:20)1 0 0(cid:21) (cid:20)0 1 0(cid:21) (cid:20)1 1 0(cid:21)
0 0 1 0 0 1 0 0 1 0 0 1
−θ θ θ θ
(cid:20)0 0 0(cid:21) (cid:20)1 0 0(cid:21) (cid:20)0 1 0(cid:21) (cid:20)1 1 0(cid:21)
0 0 0 0 0 0 0 0 0 0 0 0
−θ θ θ θ = 0
(cid:20)0 0 1(cid:21) (cid:20)1 0 1(cid:21) (cid:20)0 1 1(cid:21) (cid:20)1 1 1(cid:21)
If one writes this relation as r − r − r = 0, one obtains from this the
1 2 3
following relation between the squares θ2 εε′ :
h i
r4+r4+r4−2r2r2−2r2r2−2r2r2 = 0.
1 2 3 1 2 1 3 2 3
Replacing θ2 εε′ by θ εε′ 00 θ εε′ 01 then yields a modular form on H4
h i h i h i
of weight 8. Up to a possibly non-zero constant this is equal to F. We
can consider F as a section of the Q-line bundle L both on AIgu and AVor.
4 4
We conclude this note with a remark on the divisor defined by F, which in
the case of the Igusa compactification is essentially nothing but Mumford’s
observation.
Corollary 2.1 (i) The divisor of F on AIgu equals JIgu+DIgu.
4 4 4
(ii) The divisor of F on AVor equals JVor +DVor+4E.
4 4 4
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Proof. The divisor defined by F on A′ equals J′ +D′ ∼ 8L. (This gives
4 4 4
Mumford’s claim that J′ = 8L − D′.) Since A′ and AIgu only differ in
4 4 4 4
Igu
codimension 2 the analogous statement is true on A . Moreover, since
4
π∗DIgu = DVor +4E the form F, considered as a form on AVor vanishes at
4 4 4
least on JVor +DVor +4E. On the other hand we know that the class of
4 4
JVor equals 8L−DVor −4E by theorem 1.1 and hence F vanishes on E of
4 4
order exactly 4. 2
Remark It should be possible to compute the vanishing order of F on E
from the Fourier expansion of F with respect to E, but this seems to be a
cumbersome computation.
References
[A] V. Alexeev, Compactified Jacobians alg-geom/9608012.
[HS] K.Hulek,G.K.Sankaran,Thenefconeoftoroidalcompactifications
of A . mathAG 0203238
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[I] J.I. Igusa, Modular forms and projective invariants. Amer. J.Math.
89 (1967), 817–855.
[Mu] D. Mumford, On the Kodaira dimension of the Siegel modular vari-
ety. In: Algebraic geometry – open probems, Proceedings, Ravello
1982, eds.E.Ciliberto,F.GhioneandF.Orecchia.LectureNotes in
Mathematics. 997 Springer Verlag, Berlin, New York (1983), 348–
375.
[N] Y. Namikawa, A new compactification of the Siegel space and de-
generation of abelian varieties I, II. Math. Ann 221 (1976), 97–141,
Math. Ann. 221 (1976), 201–241.
[vG1] B. v. Geemen, Siegel modular forms vanishing on the moduli space
of curves. Invent. Math. 78 (1984), 329–349.
[vGvG] B. v. Geemen, G. van der Geer, Kummer varieties and the moduli
space of abelian varieties. Amer. J. Math. 108 (1986), 615–642.
[W] R. Weissauer, Stabile Modulformen und Eisensteinreihen. Lecture
Notes in Mathematics 1219 (1986).
Joe Harris
Department of Mathematics
Harvard University
One Oxford Street
Cambridge, MA 02138
5
USA
[email protected]
Klaus Hulek
Institut fu¨r Mathematik
Universita¨t Hannover
Postfach 6009
D 30060 Hannover
Germany
[email protected]
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