arXiv:1302.5946v1 [math.AG] 24 Feb 2013

THE SCHOTTKY PROBLEM IN GENUS FIVECHARLES SIEGEL
Abstract. In this paper, we present a solution to the Schottky problem in thespirit of Schottky and Jung for genus five curves. To do so, we exploit natural
incidence structures on the fibers of several maps to reduce all questions tostatements about the Prym map for genus six curves. This allows us to findall components of the big Schottky locus and thus, to show that the small
Schottky locus introduced by Donagi is irreducible.

Contents
Introduction
1. Background
2. Line Configurations
3. Degenerations of Abelian varieties
4. Contracted loci5. Schottky-Jung locus
References

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Introduction
The Schottky problem has a long history and is one of the first questions to be
asked in complex curve theory: how can we characterize Jacobians among all abelianvarieties? The question turns out to be quite subtle, and a completely satisfactory
answer is still lacking, despite several characterizations existing and being a fertile
ground for new techniques in studying the moduli of abelian varieties.
Historical Overview. The oldest approach to the Schottky problem is via theta
functions. Certain theta functions give natural coordinates on Ag , called thetanulls,
and Schottky [Sch88] for genus four and Schottky and Jung [SJ09] for genus g ≥5 conjectured equations for the locus of Jacobians. We denote the locus theseequations cut out by Sg , the Schottky locus, and then the Schottky-Jung conjectureis that Sg = Jg . Schottky and Jung’s results implied only Jg ⊂ Sg , though that Jg
is an irreducible component of Sg was shown by [Gee84]. A solution in the genus fourcase was announced in the late 1960s by Igusa [Igu69] and appeared independently
in [Igu81] and [Fre83]. The method was to prove that S4 was an irreducible divisor
on A4 , and thus, must be identical with J4 , an irreducible divisor contained inside
it. Unfortunately, this method cannot be employed directly in higher genus, as it
relied on the fact that S4 is given by a single equation in theta nulls.
Another attempt to describe the Jacobians involves the dimension of the singularlocus of the theta divisor. The Riemann singularity theorem implies that this1

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CHARLES SIEGEL

singular locus has dimension at least g − 4, and this is a special condition on
abelian varieties. Andreotti and Mayer [AM67] studied the locus AMg of suchabelian varieties and showed that it gives a weak solution to the Schottky problem,that is, the Jacobians are an irreducible component of AMg , but other components
exist. In genus four [Bea77a] and five [Don81, Deb90] the locus AMg has been
completely described as
AM4 = θnull,4 ∪ J4 and AM5 = A1 × A4 ∪ BE ∪ J5 ,where BE is the locus of bielliptic Prym varieties and θnull,g the locus of abelianvarieties with a vanishing thetanull, with all the non-Jacobians contained in θnull,5 .
This approach has not been carried out in higher genus fully, but singularities oftheta divisors remains an active area of study [CM08, CG00, GSM07].
A third approach has been through trisecants. Fay’s trisecant formula [Fay73]
gtells us that the natural embedding of J (C)/ ± 1 → P2 −1 has a four dimensionalgfamily of trisecant lines. Welters [Wel84] proved that if A/ ± 1 → P2 −1 has a
one parameter family of trisecant lines, then it must be a Jacobian. The familyof trisecants has an infinitesimal formulation as the KP hierarchy, and Novikov
conjectured that an abelian variety is a Jacobian if and only if the corresponding
theta function satisfies the KP hierarchy. This was proved by Shiota [Shi86]. The
strongest result on trisecants is the recent proof, using difference equations, by
Krichever [Kri10] that a single trisecant on A/ ± 1 suffices to determine that an
abelian variety is a Jacobian.
This paper. In [Don87a, Don87b], Donagi showed that the original Schottky-Jung
conjecture was incorrect. In particular, he showed that C, the locus of intermediate
Jacobians of cubic threefolds, is contained in S5 . Additionally, he offered a meansof correcting this, and conjectured that nothing not accounted for by his methods
appears in the Schottky locus. However, even in genus five a proof was out of reachat the time, with some aspects of degenerations of dimension five abelian varieties
remaining unclear until Izadi’s thesis [Iza91].
In this paper, we present a proof of
Theorem. Inside of Ag , we have
J5 = Sgsmall .
The paper is organized as follows. In section 1, we will gather results from theliterature, recall definitions and set notation for the remainder of the paper. Section
2 consists of the geometry and combinatorics of point and line configurations overF2 , and describes incidence structures on the fibers of several maps defined below.
Section 3 studies degenerations of abelian varieties of dimension five, using resultsfrom Izadi’s thesis [Iza91] to complete the computation of the degree of the map
β5 begun in section 2 via incidence structures. Section 4 consists of a study ofthe contracted loci for β5 , completing our picture of this map and allowing us toproceed to section 5, where we use this to compute the Schottky locus, and provethe main theorem.
Acknowledgments. I would like to thank first of all Ron Donagi, who suggested
this problem to me and advised me through the work as my Ph.D. thesis at the University of Pennsylvania. Also, I am indebted to Sam Grushevsky, Gavril Farkas,
Angela Ortega, Tony Pantev, Elham Izadi and Angela Gibney for conversations

THE SCHOTTKY PROBLEM IN GENUS FIVE

3

clarifying aspects of the problem as well as the geometry of the moduli spaces involved in this paper. This work was supported by World Premier International
Research Center Initiative (WPI Initiative), MEXT, Japan, and also by the University of Pennsylvania
1. Background1.1. Moduli of curves and abelian varieties. First, we will describe the various
moduli spaces that we will need, all of which are covers of the spaces Mg and Ag , themoduli space of genus g compact Riemann surfaces and of g dimensional principally
polarized abelian varieties, respectively.
We will need first Hg , the Siegel upper half space of g × g symmetric complex
matrices with positive definite imaginary part. This space is the stack universal
cover of Ag , and has an action of Γg = Sp(2g, Z) by
A B· Ω = (AΩ + B)(CΩ + D)−1
C D
such that Ag ∼= Hg /Γg as complex analytic spaces.
Other space we will need are defined by subgroups of Γg . We set Ag (n) to be
the quotient of Hg by
Γg (n) = {γ ∈ Γg |γ ≡ 1

mod n}.

This space parameterizes pairs (A, φ) where A ∈ Ag and φ : A[n] → (Z/nZ)2g is a
symplectic isomorphism from the points of order n on A to (Z/nZ)2g . For theta
functions, it will be essential to use Ag (n, 2n), which is the quotient by



A B
Γg (n, 2n) =∈ Γg (n) diag(t AC) ≡ diag(t BD) ≡ 0 mod 2n .
C D
For the Prym map, we will need to fix the vector v = (0, . . . , 0, 1/2) ∈ Q2g , andthen look at the groupsRΓg (n) =RΓg (n, 2n) ={γ ∈ Γg (n)|γ · v ≡ v

mod (nZ2g )}{γ ∈ Γg (n, 2n)|γ · v ≡ v

mod (nZ2g )}corresponding to the moduli spaces RAg (n) and RAg (n, 2n). These spaces parameterize pairs (A, µ) where A is in Ag (n) or Ag (n, 2n), respectively, along withµ ∈ A[2] nonzero.
For Mg , we will need all of the analogous spaces to the ones defined above for Ag .
They can all be constructed via pullbacks: we use the Torelli map Jg : Mg → Ag
and take the fiber product with the projection to Ag . For example, RMg is thespace of pairs (C, µ) where µ is a nonzero point of order 2 on J (C). We will need one
additional space, R2 Mg , whose elements are ordered triples (C, µ, ν) with C ∈ Mg ,µ, ν distinct nonzero points of order 2 on the Jacobian such that for any line bundle
L on C with L⊗2 ∼= KC , we have(1)h0 (C, L) + h0 (C, L ⊗ µ) + h0 (C, L ⊗ ν) + h0 (C, L ⊗ µ ⊗ ν) ≡ 0

mod 2.

The left hand side is called the Weil pairing and we will refer to pairs satisfying 1
as orthogonal. For abelian varieties that are not Jacobians, we can define the Weilpairing as the intersection pairing on H1 (A, Z/2Z). (That these are the same isshown in [Mum71])

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CHARLES SIEGEL

Finally, here we note another interpretation of RMg . The data of a smooth
curve C and µ ∈ J (C)[2] is equivalent to C̃ → C, an unramified double cover. We
can construct C̃ inside the total space of µ by fixing ϕ : µ⊗2 → OC an isomorphism
and then, on sufficiently small open sets U ⊂ C, we haveC̃|U = {σ 2 = 1|σ ∈ µ(U )}.
We will use (C, µ) and (C, C̃) interchangeably, along with (C̃, ι) where ι : C̃ → C̃is the involution corresponding to the double cover.
1.2. Compactifications. In this paper, we will need several partial compactifications of the moduli spaces we study, particularly Mg , Ag , RMg , RAg and R2 Mg .
We will start with the boundary for Ag and RAg , which is simpler than that ofMg and its covers. The space Ag has a distinguished compactification, the Satakecompactification, which all other compactifications map to. We will only need thesscorank 1 part, which we will denote by Ag and we note that ∂Ag ∼= Ag−1 . There isa larger class of compactifications called toroidal compactifications [AMRT10]. We
will only need the corank 1 part here as well, and for all toroidal compactifications,tstthis is the same, so we write Ag = BlAg−1 Ag . This has the property that ∂Ag is adivisor, and is isomorphic to Xg−1 , the universal Kummer variety over Ag−1 . We
will interpret it as parameterizing C× -extensions of abelian varieties of dimensiong − 1.
The space RAg has a slightly more complex boundary, and for reference we
reproduce results in [Don87a] describing it. There are both Satake and toroidal
compactifications, for which we will again only need the corank 1 part. In either
case, there are three components, depending on the relationship between the pointof order two, µ, and the vanishing cycle δ taken modulo two:st
• ∂ I RA ∼= Ag−1 and ∂ I RA ∼= Xg−1 are the components where δ = µ.
g

g

s
t
• ∂ II RAg ∼= RAg−1 and ∂ II RAg ∼= RX g−1 (The fiber of RX g−1 → RAg−1
over (X, µ) ∈ RAg−1 is the double cover of X determined by (µ)⊥ ) are thecomponents where δ 6= µ and the two points are orthogonal.
st• ∂ III RAg ∼= Ag−1 and ∂ III RAg ∼= Xg−1 are the components where δ andµ are not orthogonal.
In fact, we have

Proposition 1.1 ([Don87a, Proposition 2.3.2]).
tst
The projection RAg → RAg is simply ramified along ∂ III RAg and unramified onts

the other boundary components. (Note that the map ∂ III RAg → ∂ III RAg is given
×2

by RX g−1 → RX g−1 → RAg−1 with the first map multiplication by two along thefibers)For Mg and its covers, the boundary is somewhat more complex. We will only
be using partial compactifications contained in the Deligne-Mumford stable curve
compactification [DM69]. In this compactification, we have δi for 1 ≤ i ≤ g−1
2
consisting of reducible curves with a component of genus i meeting a component ofgenus g − i at a point. There is also a component δ0 consisting of irreducible curvesof geometric genus g − 1 with a single node[HM98].
The situation for RMg is slightly more complex. Each boundary component of
Mg splits into three components. The reducible components δi give us ∂i , ∂g−iTHE SCHOTTKY PROBLEM IN GENUS FIVE

5and ∂i,g−i , consisting of pairs (C, µ) where µ is supported on the curve of genus i,sg − i or both. For δ0 , we have ∂ I , ∂ II and ∂ III as with RAg . We also, for later
reference, describe the corresponding double covers:
Proposition 1.2 ([Don92, Examples 1.9]).
The generic element of the boundary components of RMg correspond to double
covers as follows:• ∂i for 1 ≤ i ≤ g − 1: the base curve is C = Ci ∪p Cg−i and the double coveris C̃ = Cg−i ∪p1 C̃i ∪p2 Cg−i .
: the base curve is C = Ci ∪p∼q Cg−i and C̃ =• ∂i,g−i for 1 ≤ g−1
2
C̃g−i ∪p1 ∼q1 ,p2 ∼q2 C̃i .
• ∂ I : the base curve is C = X/p ∼ q and the double cover is`C̃ = X0 X1 /p0 ∼ q1 , p1 ∼ q0 . This is called a Wirtinger double cover.
• ∂ II : the base curve is C = X/p ∼ q and the double cover isC̃ = X̃/p0 ∼ q0 , p1 ∼ q1 where X̃ → X is an unramified double cover. Thisis called an unallowable double cover.
• ∂ III : the base curve is C = X/p ∼ q and the double cover is C̃ = X̃/p̃ ∼ q̃
where X̃ → X is a double cover ramified over p and q. This is called a
Beauville double cover.
1.3. The Prym map. In this section, we will define, describe and collect useful
results about the Prym map.
Let C be a genus g curve and π : C̃ → C be any morphism of curves.
P Then we
can define a map Nm
:
J
(
C̃)
→
J
(C)
by
writing
D
∈
J
(
C̃)
as
D
=
nP P andPπsetting Nmπ (D) = nP π(P ). If π is surjective, then so is Nmπ . This is especially
useful in the case where π is an unramified double cover. In this case, the kernel isg − 1 dimensional and has two components. We define Pg (C, C̃) = ker0 Nmπ , theconnected component of the identity of the kernel of the norm map, and call it thePrym variety of the double cover. The principal polarization on J (C̃) restricts totwice a principal polarization on P(C, C̃)[Mum74, Corollary 2], thus giving P(C, C̃)
the natural structure of a principally polarized abelian variety, giving us a map
Pg : RMg → Ag−1 . Additionally, we have:
Lemma 1.3 (Mumford Sequence[Mum74, Corollary 1]).
Let (C, µ) ∈ RMg . Then we have a short exact sequence
0 → hµi → µ⊥ → P(C, µ)[2] → 0
where we are taking the orthogonal complement with respect to the Weil pairing.
If ν ∈ µ⊥ we will denote the image of ν in P(C, µ)[2] by ν̄.
This allows us to define also RP g : R2 Mg → RAg−1 by
RP g (C, µ, ν) = (P(C, µ), ν).
Unfortunately, as defined, the Prym map is not proper, so we must study thePrym map on the boundary of RMg .
Proposition 1.4 ([Don92, Example 1.9]).
The Prym map extends to the general point of the boundary of RMg as follows
along each component:• ∂ I : For a Wirtinger double cover, Pg (C, C̃) ∼= Jg−1 (X).

6

CHARLES SIEGEL• ∂ II : For an unallowable double cover, Pg (C, C̃) is the C× -extension of
Pg−1 (X, X̃) given by p0 − p1 + q0 − q1 in Pg−1 (X, X̃)• ∂ III : The Prym of a Beauville double cover is a principally polarized abelianvariety, and this and other ramified Prym maps are studied in [MP12]. It
is isogenous to Pg−1 (X, X̃) but is not isomorphic to it.
• ∂i fits into a diagram:Pg// Ai−1 × Ag−i
77
♦♦♦
♦
♦
♦♦♦×J
♦♦P

♦♦♦ i g−i
RMi × Mg−i∂i• ∂i,g−i fits into a diagram:∂i,g−i
RMi × RMg−i

Pg// ∂Atg−1

Pi ×Pg−i// Ak−1 × Ag−k−1

Unfortunately, Pg is only a rational map, it cannot be extended to all over RMg .
However, if we look at Pg−1 (Ag−1 ) = (RMg )allowable , then we have
Theorem 1.5 (Prym is Proper [DS81, Theorem 1.1], [Bea77a, Proposition 6.3]).
For all g, (RMg )allowable → Ag−1 is proper and for g ≤ 6 it is surjective.
Finally, we will need a theorem of Shokurov, based on Mumford’s analysis of thesingularities of theta divisors:
Proposition 1.6 ([Sho82], stated as it appears in [Bea89]).
Assume g 6= 4. If Pg (C, C̃) is a Jacobian, then C is hyperelliptic, trigonal or is aplane quintic with an even point of order two.
In particular, this theorem implies that C is not bielliptic, that is, it is not a
branched cover of an elliptic curve, and this will be the main use we will have forit later.
1.4. Fibers of the Prym map. Any study of the fibers of the Prym map begins
with the tetragonal construction of Donagi, announced in [Don81] and more fully
developed in [Don92]. Let π : C̃ → C be an unramified double cover and f : C → P1
be a degree four map. Then we definef∗ C̃ = {D ∈ Sym4 C̃| Nmπ (D) = f −1 (k) for some k ∈ P1 .
The involution ι on C̃ extends to f∗ C̃, which also splits into two components, giving
f1
f0
π
π
us two new towers C̃0 →0 C0 → P1 and C̃1 →1 C1 → P1 where (Ci , C̃i ) ∈ RMg andfi is of degree four. The key theorem is
Theorem 1.7 ([Don81, Proposition 1.1]).
The tetragonal construction commutes with the Prym map:
Pg (C, C̃) ∼= Pg (C0 , C̃0 ) ∼= Pg (C1 , C̃1 ).
This is a generalization of a theorem of Recillas identifying trigonal Pryms:

THE SCHOTTKY PROBLEM IN GENUS FIVE

7

Theorem 1.8 (The Trigonal Construction [Don92, Example 2.15(1)],[Rec74]).
If C → P1 is tetragonal and C̃ → C is the reducible double cover, then the tetragonal
construction gives a copy of C̃ → C → P1 and also P1 ∪ T̃ ∪ P1 → P1 ∪ T → P1
where T is a trigonal curve and Pg (T, T̃ ) ∼= Jg−1 (C). Additionally, this operation
is a bijection, so every trigonal Prym is a tetragonal Jacobian.
and also we recall a theorem of Mumford’s identifying hyperelliptic Pryms:
Proposition 1.9 ([Mum71]).
The Pryms of hyperelliptic curves are products of Jacobians.
Now we will discuss the fibers of the Prym maps P6 and P5 , as well as the
behavior over certain special loci. For a general genus six curve, there are exactly
five g41 ’s, and from this and the fact that any two points in the fiber of P6 are
related by at most two applications of the tetragonal construction, we get
Theorem 1.10 ([DS81, Theorem 2.1][Don81, Theorem 2.1]).
The map RM6 → A5 is generically finite of degree 27 with Galois group WE6 , theWeil group of the root system of type E6 , giving the general fiber the structure of
the 27 lines on a cubic surface.
We can see this structure clearly over some special loci in A5 . However, first
we need to perform some blowups. Let RQ0 and RQ1 be the loci of plane quintic
curves with an odd or even point of order two, respectively. (If µ is a point of order
two on J (Q), its parity is the parity of the dimension of the space of global sectionsof µ ⊗ g52 .) Also let RMT6 rig be the locus of genus six curves with a g31 , let C be the] 6 to belocus of intermediate Jacobians of cubic threefolds in A5 . Finally, set RM
the blowup of RM6 along RQ0 ∪ RQ1 ∪ RMT6 rig and Ae5 the blowup of A5 along] 6 → Ae5 . This blown up Prym map is easierJ5 ∪ C. Then the Prym map P̃6 : RM
to analyze and gives more explicitly the structure of the tetragonal construction
and the fibers.
Proposition 1.11 ([Don92, Remark 4.5.1]).
Any two points in a fiber of P̃6 are related by at most two tetragonal constructions.
Proposition 1.12 ([Don92, 4.3.4]).
The fiber of P6 over X ∈ C is the Fano surface of lines in X. In Ae5 , the inverse
image of X ∈ C is pairs (X, H) where H is a hyperplane in P4 . Then, P̃6−1 (X, H) ={ℓ|ℓ ⊂ X ∩ H}.
The above proposition exhibits the structure of the 27 lines very explicitly, andit follows from [Bea77b, Theorem 2.1] which uses the correspondence between conic
bundle structures and lines on a cubic threefold to identify the intermediate Jacobians with the Pryms of discriminant curves.
Suppressing normal data, we can also describe the fiber over the Jacobian of acurve:
Proposition 1.13 ([Don92, 4.3.7]).
Let C ∈ M5 . Then P̃6−1 (J (C)) contains one plane quintic with an odd pointof order two, ten trigonal curves with double covers and sixteen Wirtinger double
covers.
Another extremely useful result is

8

CHARLES SIEGEL

Proposition 1.14 ([Don81, Corollary 2.3],[Don92, 4.8],[Cle83]).
The ramification locus of P6 is mapped six to one onto the branch locus, which is
the locus of intermediate Jacobians of quartic double solids.
Finally, for P6 we look over the boundary where we have
Proposition 1.15 ([Iza91, Corollary 6.5]).
t
The restriction of the Prym map to ∂ II → ∂A5 is generically unramified and finite.
Now, we recall the major results on P5 :Proposition 1.16 ([Don92, Theorem 5.2]).
(1) There exists an involution λ : RM5 → RM5 such that P5 ◦ λ = P5 .
t00(2) There is a natural birational map χ : A4 99K RC , where RC is the spaceof cubic threefolds with an even point of order two on the intermediate
Jacobian, including nodal cubic threefolds as limits.
Given these two maps, Donagi showed
Proposition 1.17 ([Don81, Theorem 3.3]).
For A ∈ A4 generic, P5−1 (A)/λ is isomorphic to the Fano surface of lines in χ(A),and (C1 , C̃1 ) and (C2 , C̃2 ) are tetragonally related if and only if the correspondinglines intersect. Thus, any two points in P5−1 (A) are related by at most two tetragonal
constructions.
Before moving on, we will identify the fibers of P5 over Jacobians:
Proposition 1.18 ([Don92, Theorem 5.14]).
Let B ∈ M4 be a general curve of genus 4 and let (X, δ) = χ(J (B)). Then
(1) X is a nodal cubic threefold(2) The double cover of the Fano surface of lines in X is reducible and eachcomponent is isomorphic to Sym2 B(3) P5−1 (J (B)) is isomorphic to the double cover of the Fano surface of lines ofX, with one component of trigonal curves Tp,q and one of Wirtinger double
covers Sp,q , for (p, q) ∈ Sym2 B(4) The tetragonal construction takes Sp,q and Tp,q to Sr,s and Tr,s if and only
if p + q + r + s is a special divisor on B, and λ exchanges Sp,q and Tp,q(5) Two objects of P5−1 (J (B)) are related by at most two tetragonal constructions.
1.5. Theta Functions. Now, we shall describe one more major technical tool usedin this paper. The Riemann theta function on Hg × Cg is given byX
θ(Ω, z) =exp[πi(t nΩn + 2t nz)].
n∈Zg

This function is periodic with respect to Zg and is multiplied by an exponential
factor with respect to ΩZg . Thus, the zero locus of θ(Ω, z) is periodic for Zg ⊕ ΩZg
and gives a divisor Θ on A, the theta divisor.
For ǫ, δ ∈ Qg , we define a theta function with characteristics to be ǫθ(Ω, z) = exp[πi(t ǫΩǫ + 2t ǫ(z + δ))]θ(Ω, z + Ωǫ + δ)δTHE SCHOTTKY PROBLEM IN GENUS FIVE

9

which is essentially the translate of θ(Ω, z) by Ωǫ + δ. Evaluating at z = 0, theseare Siegel modular forms of weight 21 and level (4, 8), so they are only well-definedon Ag (4, 8), not on Ag itself.
 ǫ
We will also need θ2 [ǫ](Ω, z) = θ(2Ω, 0) where ǫ ∈ ( 12 Z/Z)g , the second or0ǫ0
der theta functions, and θ(2Ω, 0) where ǫ ∈ ( 12 Z/Z)g−1 . These are mod0 1/2
ular forms of weight (2, 4), and we we can use them to define maps αg : Ag (2, 4) →P(Ug ) and βg : RAg (2, 4) → P(Ug−1 ) where Ug is the vector
 space of functions
ǫ
0(Z/2Z)g → C by setting αg (Ω)ǫ = θ2 [ǫ](Ω, 0) and βg (Ω)ǫ = θ(2Ω, 0).
0 1/2
We get maps on Ag and RAg by noting that Gg = Γg /Γg (2, 4) acts on Ag (2, 4),
RAg (2, 4) and P(Ug ) in compatible ways, so that if Pg = P(Ug )/Gg , we have
αg : Ag → Pg and βg : RAg → Pg−1 .
The primary purpose of this paper will be to understand the fibers of β5 . For
this, we will use
Theorem 1.19 (Theta symmetry [Don87b, Theorem 3.1]).
Let C ∈ Mg+1 be a curve of genus g + 1 and let {0, µ0 , µ1 , µ2 } be a rank 2 isotropic
subgroup of Jg+1 (C)2 (thus, µ2 = µ0 + µ1 ). For i = 0, 1, 2 we have a Prym variety
Pi = P (C, µi ) ∈ Ag and on it a uniquely determined semiperiod νi , the image ofµj , j 6= i in Pi .
The point β(Pi , νi ) is independent of i = 0, 1, 2.
We will also use the description of the fibers of β4 that appears in the survey[Don88]Proposition 1.20 ([Don88, Theorem 5.3]).
For C ∈ M3 , the fiber β4−1 (α3 (J (C))) consists of two copies of the Kummer
K(J (C)), one contained in the interior of RM4 and the other the fiber oversts
J (C) ∈ A3 = ∂A4 in the projection A4 → A4 .
1.6. The Schottky Loci. We begin with the theorem that motivates this study:
Theorem 1.21 (Schottky-Jung Identities [Sch88, SJ09, RF74]).
The following diagram commutes:
RMg⑧ ❄❄❄ RJ
⑧
❄❄ g
⑧
⑧
❄❄
⑧⑧
❄
⑧
⑧
Ag−1
RAg
❄❄
⑧
❄❄
⑧⑧
❄
⑧⑧β
αg−1 ❄❄⑧❄⑧⑧ gPg
Pg

Schottky and Jung noticed these relations between various theta functions andsuggested that they can be used to describe Jacobians. More precisely, we define
RS g= βg−1 (im αg−1 )Sgbig= {A|∃µ, (A, µ) ∈ RS g }Sgsmall= {A|∀µ, (A, µ) ∈ RS g }10

CHARLES SIEGEL

Schottky and Jung conjectured that Sgbig = Jg , the closure of the image of theTorelli map. This is known not to be true, and the key result for that is Theta
symmetry (Theorem 1.19)
Let Q ⊂ P2 be a plane quintic curve and let µ, ν be two points of order two on
J (Q) such that µ is odd and ν is even. Then (P(Q, µ), ν̄) ∈ RS 5 if and only if(P(Q, ν), µ̄) ∈ RS 5 . However, as mentioned above and proved in [DS81], P(Q, µ)
is the intermediate Jacobian of a cubic threefold. Additionally, P(C, ν) is known
to be a Jacobian of a curve (see the discussion in section 5 of [Don87b]). Thus,
C ⊂ S5big , and so Donagi introduced Sgsmall to correct this, as only RC 0 appears in
RS 5 , not RC 1 , the locus of intermediate Jacobians of cubic threefolds with an odd
point of order two.
There are two additional components that are not difficult to show lie in RS 5 .
There is the locus RA1 × A4 , and in fact:
Proposition 1.22 ([Don87a, 3.3.4]).
The Schottky locus RS g contains Ag′ ×RAg′′ for all g ′ +g ′′ = g (Note that if g = 0,then RS 0 = ∅).
We will see below that no other product loci can be components of RS 5 .
There is also one boundary component, described by
Proposition 1.23 ([Don87a, Theorem 3.3.1]).
In the Satake compactification, we havesss

∂RS g = ∂ I RAg ∪ ∂ III RAg ∪ iII (RS g−1 )swhere iII is the inclusion of RAg−1 as ∂ II RAg .
t

In [Don88] this result is strengthened to show that ∂RAg does not contain ∂ III
or iII (RS g−1 ) as components, but only contains the points that are limits of othercomponents.
2. Line Configurations
In this section, we will study a class of configurations that occur whenever there
is a triality on a space, such as the tetragonal construction or theta symmetry.
Because we are working with trialities, everything in this section will be done overF2 , but the majority will work for Fq .
Definition 2.1 (Line Configuration).
A line configuration V over F2 is a set PV , called the points of V , along with a set
LV , called the lines of V , such that for all ℓ ∈ LV , we have ℓ ⊂ PV , |ℓ| = 3 and for
all ℓ, ℓ′ ∈ LV , |ℓ ∩ ℓ′ | ≥ 2 implies that ℓ = ℓ′ .
We can construct a large class of examples, which we will call algebraic line
configurations. These are defined by starting with V ⊂ Pn over F2 a projective
variety. Then we set PV to be the F2 -points of V and LV to be the projective lines
over F2 contained in V . The first interesting example is V = P2 , the Fano plane:
•
☛ ✸✸✸
☛
☛
•
•
✟ • ❖ ✻✻✟♦✟ ♦♦♦ ❖❖❖✻•••

THE SCHOTTKY PROBLEM IN GENUS FIVE

11

we define a morphism of configurations f : V → W to be an injection f : PV →PW such that for each line {p0 , p1 , p∞ } ∈ LV , the image {f (p0 ), f (p1 ), f (p∞ )} is aline in LW . A subconfiguration is then the image of a morphism. We say that two
points are collinear if they lie on a subconfiguration isomorphic to P1 and that twolines are coplanar if they lie on a subconfiguration isomorphic to P2 .
Definition 2.2 (V -Configuration).
Let V be a line configuration. Then another line configuration W is a
V -configuration if for each p ∈ PW , there exists a bijection φp : {ℓ ∈ LW |p ∈ ℓ} →PV such that ℓ, ℓ′ are coplanar if and only if φp (ℓ) and φp (ℓ′ ) are collinear.
Example 2.3.
(1) Pn is a Pn−1 -configuration.
(2) If V is a collection of n points with LV = ∅, then (P1 )n is a V -configuration.
(3) For some V , we can construct examples of multiple fundamentally distinct
V -configurations. For instance, if V consists of five points and no lines,
the previous example says that (P1 )5 is a V -configuration. However, if
S is a smooth cubic surface, then we can define a configuration W with
PW = {ℓ ⊂ S|ℓ is a line} and with LW the set of triples of coplanar lines.
It is classical that given a line on a cubic surface, it is contained in exactly
five coplanar triples, no set of which are configured as a Fano plane.
Now, to help us to describe the properties of line configurations, we use theincidence graph. This is the graph ΓV whose vertices are the points of PV and two
vertices p, q ∈ PV are connected by an edge if and only if they are collinear. We
fix a metric on ΓV such that each edge has length 1, and give properties to V fromthe properties of the metrized graph ΓV , for instance, we can speak of connectedconfigurations or the diameter of a configuration.
Given ΓV , we can define numerical invariants of a configuration. For each p ∈ PV ,we define Vi (p) = {q ∈ V |d(p, q) = i}, the points that are first reached after passing
along i lines from p. For points p ∈ PV , and q ∈ Vi (p), we define Vi,j (p, q) =Vj (p) ∩ V1 (q), the points distance j from p which are adjacent to q. Note that
Vi,j (p, q) is empty unless j is i − 1, i or i + 1. We denote the cardinalities of thesesets by vi (p) and vi,j (p, q) (by convention, we set these numbers to be zero if i or j
is negative), and call a line configuration symmetric if the vi (p) and vi,j (p, q) don’t
depend on p and q, in which case we will denote vi = vi (p) and vi,j = vi,j (p, q).
Proposition 2.4.
Let V be a connected symmetric line configuration. Then
(1) For all i, v1 = vi,i−1 + vi,i + vi,i+1(2) For all i, v1 vi = vi−1,i vi−1 + vi,i vi + vi+1,i vi+1
(3) We have v0 = 1, v0,0 = 0, v0,1 = v1 and v1,0 = 1.
Let W be a connected symmetric V -configuration. Then
(4) w2,2 ≥ w2,1
(5) w1 = 2|PV |(6) w1,1 = 2v1 + 1(7) If, additionally, v3 = w3 = 0, then either w2 = |PV | or w2 = 4v2 .
For convenience, throughout the proof we will set LinesW (p) = {ℓ ∈ LW |p ∈ ℓ},
the set of lines through p in W .

12

CHARLES SIEGELProof. Let V be a connected symmetric line configuration and W a connected
symmetric V -configuration.
(1) Fix p ∈ V , i ∈ N, q ∈ Vi (p). Then
Vi,i−1 (p, q) ∪ Vi,i (p, q) ∪ Vi,i+1 (p, q)= (Vi−1 (p) ∩ V1 (q)) ∪ (Vi (p) ∩ V1 (q)) ∪ (Vi+1 (p) ∩ V1 (q))= (Vi−1 (p) ∪ Vi (p) ∪ Vi+1 (p)) ∩ V1 (q)= V1 (q).
(2) Fix p ∈ V , i ∈ N. Let X = {(a, b)|a ∈ Vi (p), b ∈ V1 (a)}. Then |X| = vi v1 .
But also,= {(a, b)|a ∈ Vi (p) ∩ V1 (b), b ∈ ∪j Vj (p)}= ∪j {(a, b)|a ∈ Vj,i (p, b), b ∈ Vj (p)}P
and so |X| = j vj,i vj .
(3) These all follow directly from the definitions.
(4) Fix q ∈ W2 (p). We prove that no line containing q contains two points ofW1 (p). As each line consists of 3 points, this implies that w2,2 ≥ w2,1 . Leta, b ∈ W1 (p) and assume that there is a line ℓ ∈ LW such that a, b, q ∈ ℓ. Asa, b ∈ W1 (p), there exist lines m1 , m2 through p containing a, b respectively.
But then, m1 , m2 must be coplanar, and so there is a line m containing p, q,
so q ∈ V1 (p) ∩ V2 (p) = ∅, a contradiction.
(5) For each line ℓ ∈ LinesW (p), fix a bijection`ℓ → P1 (F2 ) such that p ismapped to ∞. Then ∪ℓ∈LinesW (p) ℓ \ {p} = ℓ∈LinesW (p) A1 (F2 ), and thishas cardinality twice the number of lines, 2|PV |.
(6) Fix p ∈ PW , ℓ a line through p, p′ ∈ ℓ distinct form p. Through p′ , there
are |PV | lines. One is ℓ, v1 of them are coplanar with ℓ, and the rest arenot. Each coplanar line consists of n points in V1 (p) that are not p′ , butthere are also n − 1 points of ℓ in V1 (p) other than p′ , and so w1,1 = 2v1 + 1.
(7) Fix p ∈ W . Let X̃ be the set of triples (ℓ, m, q) in LinesW (p) × LW × W2
such that p ∈ ℓ, ℓ ∩ m 6= ∅ and q ∈ m. There is a natural map X̃ → W2
which is surjective. Then
X

|X̃| ==

w2 · |fiber|
w2 · |{paths to q ∈ W2 from q}|==

w2 · |{points ofW1 (p)connected to p}|
w2 · |W1 (p) ∩ W1 (q)| for q ∈ W2 (p)=w2 w2,1`
Now, we also have a map X̃ → LinesW (p) × W2 (p) = A B, where
the fiber over A has cardinality 1 and over B has cardinality 0. These are
the only possibilities, because if there were two, then we get a plane andq ∈ W1 (p). So |A| = |X̃| = w2 w2,1 , and |A| + |B| = | LinesW (p) × W2 (q)| =|PV |w2 , so |B| = w2 (|PV | − w2,1 ).
But, as W is symmetric, we can see that |V | = |PV |α, where α is thenumber of lines in LinesW (p) that don’t have a line connecting them toq. Then α = |PV | − |{lines in LinesW (p) connected to q}|, which is α =|V | − w2,1 , so |B| = |PV |(|PV | − w2,1 ).

THE SCHOTTKY PROBLEM IN GENUS FIVE

13

So, w2 (|PV | − w2,1 ) = |PV |(|PV | − w2,1 ), implying that either w2 = |PV |or else w2,1 = |PV |. In the latter case, the previous parts of this proposition
along with the hypothesis that diam(W ) ≤ 2, implies that w2 = 4v2 .
We can now apply these numerics to another family. Let the zero locus of
x21 + x22 + x1 x2 + x3 x4 + . . . + x2n−1 x2n in P2n−1 (F2 ) be denoted by Q−
2n . This is thesmooth quadric of Witt defect 1 in P2n−1 over F2 . We have actually already seen it
in an above example: if n = 2, then the zero locus is five points, no three collinear,and if n = 3, we get 27 points, each of which lies on 5 lines (and thus is adjacent to
10 points), which gives the configuration of lines on a smooth cubic surface. The
latter fact also appears as an isomorphism W E6 ∼= O6− (F2 ), the orthogonal group22
preserving the form x1 + x2 + x1 x2 + x3 x4 + x5 x6 .
−For all n, in fact, we can see that Q−
2n+2 is a Q2n -configuration, by notingthat they are homogeneous varieties, and so we only need to look at the point[0 : . . . : 0 : 1], where everything is easy to compute. Additionally, for n ≥ 3 these
are all symmetric and have diameter two, so all of the numerical conditions above
apply.
Lemma 2.5.
Let n ≥ 2. Then the only connected, symmetric Q−
2n -configuration of diameter 2 isQ−
2n+2 .
Proof. We set V = Q−
2n for some fixed n ≥ 2 and W a connected, symmetric
V -configuration.
As we are assuming diameter 2, we have to determine wi and wi,j for 0 ≤ i, j ≤ 2.
Proposition 2.4(3) determines w0 , w0,0 , w1,0 and that w0,1 = w1 . Then Propositionn−1 n
2.4(5) says that w1 = 2|Q−
(2 − 1) − 1) = 2n (2n − 1) − 2 = 22n − 2n − 2.
2n | = 2(2This leaves w1,1 , w2,1 , w1,2 , w2,2 and w2 . Proposition 2.4(6) gives w1,1 = 2(22n−2 −n−12− 2) = 22n−1 − 2n − 4. Then Proposition 2.4(1) says that 22n − 2n − 2 =1 + w1,1 + w1,2 and 22n − 2n − 2 = w2,1 + w2,2 , which we solve for w1,2 =22n − 2n − 2 − 1 − (22n−1 − 2n − 4) = 22n − 22n−1 + 1.
Substitution of 2.4(1) into 2.4(2) and cancellation gives us the relation wi wj,i =wj wi,j . And thus, w1 w2,1 = w2 w1,2 . So (22n−1 − 2n − 4)w2,1 = w2 (22n − 2n −2 − 1 − (22n−1 − 2n − 4)). Using 2.4(7), we know that w2 = (2n−1 (2n − 1) − 1) or4((2n−1 (2n − 1) − 1) − 1 − 2((2n−2(2n−1 − 1) − 1))) = 22n . The former case leads tow2,2 < w2,1 contradicting 2.4(4), and so we must have the latter case. And finally,
this determines w2,1 using w1 w2,1 = w2 w1,2 and w2,2 from w1 = w2,1 + w2,2 .
n n+1
In particular, |W | = |Q−− 1) − 1.
2n+2 | = 2 (2n n+1
We now prove that these 2 (2− 1) − 1 points must be connected by lines ina unique way to satisfy the conditions of the lemma. We begin with a point of theconfiguration w ∈ W . As w2,1 = w2,2 = 2n−1 (2n − 1) − 1 = |Q−
2n |, each line notcontaining w must contain one point in W1 (w) and two points of W2 (w). Thus, apoint in W2 (w) is given by a point (not w) on each line containing w.
The same holds for any point w′ ∈ W1 (w), and there must be a function
W1 (w) → W1 (w′ ) that preserves incidence and takes the set of choices at w to
those at w′ . This will dictate the data of which half (we note that w1,2 = 12 w2 , andso each point of W1 (w) is colinear with half of the points of W2 (w)) of the points ofW2 (w) are connected to w′ . As this works for each w′ ∈ W1 (w), there is only one

14

CHARLES SIEGEL

such choice that will work globally, up to automorphisms of Q−
2n and involutions
switching non-w points on a line.

So to identify a Q−
2n -configuration, we must merely check symmetry and diameter.
Proposition 2.6.
The general fiber of β5 : RA5 → P15 /G4 is a Q−
6 -configuration.
Proof. Let (A, µ) ∈ RA5 be a general point. Then A is the Prym variety of 27
distinct curves (Xi , νi ) ∈ RM6 , and µ lifts to two distinct points of order two on
J (Xi ), µ0i , µ1i . Then theta symmetry implies that, again generically, (P(Xi , µji ), ν̄i )are distinct points in the fiber of β5 . Thus, (A, µ) lies on 27 triples (A, µ), (P(Xi , µ0i ), ν̄i ),(P(Xi , µ1i ), ν̄i ), which are lines over F2 , form the configuration Q−
6 , by Theorem-configuration.

1.10, and so the fiber is a Q−
6
Lemma 2.7.
The connected components of the Q−
6 -configuration on the fibers of β5 has diameter2.
g 6 , then Proposition
Proof. If we blow up RP 6 : R2 M6 → RA5 to a finite map RP
01.12 says that the fiber over (X, µ) ∈ RC , with the normal data suppressed for easeof notation, consists of 54 plane quintics with an odd and an even point of order twomarked. Theta symmetry then says that the Pryms with respect to the even points
with the images of the odd points marked are in the same fiber of β5 , and all of these
are the Jacobians of curves. By Proposition 1.13, the fiber of a Jacobian consistsof quintic curves, whose Pryms are cubic threefolds and Jacobians, trigonal curves,
whose Pryms are Jacobians, and singular curves, whose Pryms are Jacobians andtdegenerate abelian varieties in ∂ I RA5 .
If we start with a degenerate abelian variety, then we must only have singularcurves lying over it. Proposition 1.4 tells us that these are either in ∂ II or ∂i,6−i .
But the latter component only gives products, and so generically all 27 preimagesttunder P6 are in ∂ II . Thus, any points of RA5 related to a point of ∂ I RA5 is eithertin ∂ I RA5 or is a Jacobian, and so no new types of points occur after the second
iteration of theta symmetry.
Finally, we show that the objects obtained previously don’t constitute anything
new. First, the Jacobians obtained from the chosen Jacobian in the second step.
These Jacobians all must have already been obtained from the cubic threefold,because each Jacobian is the Prym of a unique quintic curve, and will thus determine
a cubic threefold. However, generically no two cubic threefolds can be related inthis manner, because the incidence is contained in the fibers of β5 , which induces
a birational isomorphism between RC 0 and α(A4 ), and so, generically, must be
injective. And last, we have the objects obtained from a given degenerate abelianvariety. We have twenty-seven nodal curves, and each occurs as a Wirtinger andas a ∂ II double cover, which we then take the Pryms of. The Wirtingers must be
Jacobians we’ve already seen, by the above, and so the ∂ II double covers would
be obtained by applying theta symmetry on those Jacobians first. Thus, the Q−
6configuration must have diameter 2.
And so, by Lemma 2.5, the map β5 must have degree a multiple of 119.

THE SCHOTTKY PROBLEM IN GENUS FIVE

15

3. Degenerations of Abelian varieties
In this chapter, we will show that the Q−
6 -configuration on the fibers of β isconnected. This amounts to computing the degree of β, which we will do by studying certain degenerations. We will spend this chapter studying the structure oft∂ II RA5 , the degenerations with vanishing cycle orthogonal but not identical to themarked semiperiod.
Lemma 3.1 ([Don87a, Corollary 3.2.3 and Lemma 3.3.6]).
tt
The extension of βg : RAg → P(Ug−1 )/Gg−1 to ∂ II RAg is βg−1 and the diagram:
RAg−1

iII

// RAt

g

βg−1

βg


P(Ug−2 )/Gg−2// P(Ug−1 )/Gg−1

is Cartesian.
tThis result tells us that if we start with a point in ∂ II RAg then the whole fibertover its image is in ∂ II RAg , and so we focus our attention there:
Proposition 3.2.
t
Any two points in a fiber of β5 inside ∂ II RA5 are related by a sequence of theta
symmetries.
Proof. By Lemma 3.1, we are working with the map β4 : RA4 → P7 /G3 . Now, fix
a curve C ∈ M3 . Then the fiber of β4 over α3 (J (C)) is Bl0 K(J (C)) ∪ K(J (C))
by Proposition 1.20, with the first component consisting of genus 4 Jacobians andt
the second of degenerate abelian varieties in ∂ I RA4 . Over (B, µ) ∈ RJ 4 , thefiber of RP 5 is a double cover of Sym2 B ∪ Sym2 B, with the first component
consisting of trigonal curves and the second consisting of Wirtinger curves, as in
Proposition 1.18. Thus, for each (B, B̃), we get a map from the double cover of2^
Sym2 B, Sym
B → Bl K(J (C)), because theta symmetry takes trigonal curves
0

to trigonal curves and Wirtinger curves to Wirtinger curves. Each of these maps
has two dimensional image, and they’re all nonisomorphic, and thus distinct, so thedimension of the union is at least three, so surjectivity follows.

Now that we have a locus where we know that theta symmetry spans the fibers,
we need to identify theta symmetry over that locus:tProposition 1.15 tells us that the fiber in ∂ II RA5 has the Q−
6 -configuration
structure we expect, and more so, the fact that theta symmetry spans the fibers
implies connectivity, so the fiber is Q−
8 . It remains, though, to check that β5 istIIgenerically unramified on ∂ RA5 .
RP

t

β5

We have a tower of maps R2 M6 →6 RA5 → P15 /G4 and as each map isgenerically finite, if we blow up each locus with positive dimensional fibers, andthe blow up again so that the maps are all well defined, the maps will be finite.
2 M → RA^e Set X =g 5 → P.
We denote the blowups and blown up maps by R
6gg
RA5 ×P RA5 . For each point (x, y) ∈ X, we can associate the distance in ΓQ− , or8
∞ if they are not connected. This gives a decomposition X = I1 ∪ I54 ∪ I64 ∪ I,

16

CHARLES SIEGEL

where I1 is the diagonal, I54 are theta related pairs, I64 the pairs theta related to
a common point and I the pairs in the same fiber but not theta related. Our goal
is to show that I = ∅.
We will focus on studying I54 . It fits into a diagram
β̃

// RA
g5
⑧??
⑧
⑧
⑧⑧
⑧
⑧⑧
⑧⑧
2:1
⑧
⑧⑧ P̃
⑧⑧
⑧
⑧⑧
 ⑧⑧
I27
I54

g 5 from A5 , and I54 → I27 is thewhere I27 is the pullback of the Prym map to RA
natural double cover. Differentiating these maps, we can see that Ram β̃ = Ram P̃.
Lemma 3.3.
Let f : X → Y be a finite morphism of smooth varieties and f¯: X ×Y X → X be
the fiber product of f with itself. Then if p is a ramification point of f and q is anonramification point in the same fiber, (p, q) is a ramification point of f¯.
Proof. Set b = f (q) = f (p). As p is a ramification point and q is not, we have dfq
an isomorphism and dfp not. Locally, this means that we have an isomorphism of
X and Y near q but not near p, and so the map df¯(p,q) : Tp,q (X ×Y X) → Tp X is
not an isomorphism.
Lemma 3.4.
g 5 . Then if β5 is ramified at p, I54 is ramified over p.
Let p ∈ ∂ II RA

g 5 be a ramification point of β̃. Then either there exists a
Proof. Let p ∈ ∂ II RA
q such that (p, q) ∈ I54 is unramified, or else not. If there is such a q, then thisfollows from Lemma 3.3.
If not, then we must have a fiber of the Prym map which is totally ramified.
This would correspond to a cubic surface such that every line has multiplicity atleast two, but which is reduced and irreducible. This cannot happen, as if there is aline of multiplicity two, then the residual intersection with the tangent plane alongthat line must be a line of multiplicity one, so every line would have multiplicity
three, which does not occur.
More useful than the lemma is the contrapositive: that if I54 is unramified overp, then β5 is unramified at p.
Proposition 3.5.
t
The map β5 is generically unramified on ∂ II RA5 .
Proof. Lemma 3.4 tells us that a point is a ramification point for β5 only if it is abranch point of β̃, which is the same as being a branch point over P̃. But Izadi’stTheorem 1.15 tells us that the Prym map is generically unramified over ∂ II RA5 ,g 5 is generically unramified over ∂ II RA
g 5 , and so, β5 is as well. and so I54 → RA
Thus, we have the main result of this section:

THE SCHOTTKY PROBLEM IN GENUS FIVE

17

Theorem 3.6.
t
the fibers of β5 : RA5 → P15 /G4 are exactly the orbits of theta symmetry.
4. Contracted lociNow, we’ve identified the fibers of β5 with the orbits of theta symmetry, which
gives us the degree of this generically finite map. What remains is to understandthe locus where β5 has infinite fibers. As each abelian variety has finitely many
points of order two, and as αg is finite and generically injective for all g [SM94,
Proposition 1], any infinite fiber must arise from an infinite fiber of the Prym map.
By Proposition 1.11 we know that given (C, µ) ∈ RM6 , we obtain the rest of
the fiber containing (C, µ) by iterating the tetragonal construction, so long as weallow points to be related via the boundary. There can be no infinite chains ofcurves with finitely many g41 ’s, because such a fiber would be countable, and an
infinite fiber must be uncountable. Thus, infinite fibers will correspond to curves
with infinitely many g41 ’s and to singular curves.
Theorem 4.1 ([ACGH85, Theorem IV.5.2],[Mum74, Theorem in Appendix]).
Let C be a smooth non-hyperelliptic curve of genus g ≥ 4. Let d, r ∈ Z with
0 < 2r ≤ d and 2 ≤ d ≤ g − 2. Then, if there exists a (d − 2r − 1)-dimensionalfamily of gdr ’s on C, we must have that C is trigonal, bielliptic or a plane quintic.
Fixing g = 6, d = 4 and r = 1, the hypotheses on Theorem 4.1 are satisfied,and so a curve of genus 6 that has a positive dimensional family of g41 ’s must be
hyperelliptic, trigonal, bielliptic, a plane quintic, or singular.
Proposition 1.9 says that Pryms of hyperelliptic curves are products. Theorem
1.8 tells us that trigonal Pryms are Jacobians. Propositions 1.12 and 1.13 tell usthat Pryms of quintics are Jacobians or cubic threefolds and Proposition 1.4 saystthat the Prym varieties of singular curves are contained in J5 , ∂A5 , the Beauville
Pryms or the product loci.
But we actually only care about contracted loci over α4 (A4 ), which may beirreducible components of RS 5 . As dim(α4 (A4 )) = 10, any irreducible component
0of RS 5 must have dimension at least 10. Also, as we will see later, RJ 5 , RC , and∂ I RA5 are the only components of RS 5 with positive local degree, thus, any othercomponent is actually a contracted locus.
Proposition 4.2.
The only component of β5−1 (α4 (A4 )) that is blown down by β is RA1 × A4 .
Proof. By the above discussion, the only loci with positive dimensional fibers arettt
RJ 5 , the bielliptic loci, RC 0 , RC 1 , ∂ I RA5 , ∂ II RA5 , ∂ III RA5 , and RP 6 (∂ III R2 M6 ).
t
The loci RJ 5 , RC 0 and ∂ I RA5 actually have positive local degree (see next
chapter) and so are not blown down.
The bielliptic loci are ruled out by 1.6.
The locus RC 1 is not in RS 5 , as any point of RC 1 will only be theta related
to other points of RC 1 . Specifically, a curve over (X, µ) ∈ RC 1 must be a plane
quintic Q and an odd point of order two ν with P (Q, ν) = X, but also the lifts ofµ must be odd, so all three points in any instance of theta symmetry give points of
RC 1 .

18

CHARLES SIEGELt

From 1.23, it is shown that ∂ III RA5 , the Beauville Pryms, RA4 × A1 andRA4 × RA1 are not in the Schottky locus, and all other product loci other thanRA1 × A4 are of too small dimension. However, RA1 × A4 is contracted by β5 .
t
The only remaining component is ∂ II RA5 , but we know this locus is mappedto P7 /G3 , and so any points of it in RS 5 must be the limits of points of othercomponents.

5. Schottky-Jung locus
Finally, we prove our main theorem:
Theorem 5.1.
tt
In RA5 , we have RS 5 = RJ 5 ∪ RC 0 ∪ ∂ I RA5 ∪ A4 × RA1 .
Proof. As the general fiber of β is a connected Q−
6 -configuration, we have thatdeg β = 119. Then Proposition 4.2 tells us that the only blown down component
of RS 5 is A4 × RA1 , so it remains to identify the components with positive local
degree.
It is known Proposition 1.16 that β5 restricts to χ−1 : RC 0 99K A4 , and thus has
local degree one.
The Schottky-Jung relations imply that the restriction to RJ 5 is birational to
the Prym map RM5 → A4 . Thus, over A ∈ A4 , we get a double cover of the Fanosurface of a cubic threefold. However, when we blow up to obtain a finite map, we
get a double cover of the 27 lines on a cubic surface, and so we have local degree
54.
t
In [GG86], van Geemen and van der Geer compute the local degree on ∂ I RA5 as
follows: fix X ∈ A4 . Then the part of the fiber of β5 is K(X) and by blowing up,
they showed that the degree of the map is the same as that of the map K(X) → P4
given by the linear system Γ00 = {s ∈ Γ(X, 2Θ)| mult0 s ≥ 4}. We blow X up at 0,
call the exceptional divisor E, and then this linear system is precisely 2Θ − 4E on
the abelian variety. Thus, the local degree is 12 (2Θ − 4E)4 = 64.
This gives us degree 1 + 54 + 64 = 119, and so there are no other components
with positive local degree.
And so, as noted in [Don87b], this implies
Corollary 5.2.
S5small = J5
Proof. The points of S5small are just the abelian varieties A such that (A, µ) ∈RS 5 for all nonzero points of order two µ ∈ A. Among the components above,
Jacobians have all nonzero points of order two, but the intermediate Jacobians of
cubic threefolds only have even points of order two, the boundary component has
only the vanishing cycle, and the product locus only has pullbacks of the points oforder two on the elliptic curve.
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Kavli IPMU (WPI), The University of Tokyo, Kashiwa, Chiba 277-8583, Japan
E-mail address: charles.siegel@ipmu.jp