Andreotti-Mayer loci and the Schottky problem

0.5 MB·English

by  Ciro Ciliberto

ANDREOTTI–MAYER LOCI AND THE SCHOTTKY PROBLEM
7
0
CIROCILIBERTOANDGERARDVANDERGEER
0
2
Abstract. We prove a lower bound for the codimension of the Andreotti-
n
a Mayer locus Ng,1 and show that the lower bound is reached only for the hy-
J perellipticlocusingenus4andtheJacobianlocusingenus5. Inrelationwith
theintersectionoftheAndreotti-Mayerlociwiththeboundaryofthemoduli
2
spaceAg westudysubvarietiesofprincipallypolarizedabelianvarieties(B,Ξ)
1
parametrizing points b such that Ξ and the translate Ξb are tangentially de-
generate alongavarietyofagivendimension.
]
G
A
.
h 1. Introduction
t
a TheSchottkyproblemasksforacharacterizationofJacobianvarietiesamongall
m
principally polarized abelian varieties. In other words, it asks for a description of
[ theJacobianlocus inthemodulispace ofallprincipallypolarizedabelianva-
Jg Ag
rieties of givendimension g. In the 1960’sAndreotti and Mayer (see [2]) pioneered
1
v an approach based on the fact that the Jacobian variety of a non-hyperelliptic
3 (resp. hyperelliptic) curve of genus g 3 has a singular locus of dimension g 4
5 (resp. g 3). They introduced the loc≥i N of principally polarized abelian v−ari-
g,k
3 −
eties (X,Θ ) of dimension g with a singular locus of Θ of dimension k and
1 X X ≥
showed that (resp. the hyperelliptic locus ) is an irreducible component of
0 Jg Hg
7 Ng,g−4 (resp. Ng,g−3). However, in general there are more irreducible components
0 ofN sothatthedimensionofthesingularlocusofΘ doesnotsufficetochar-
g,g−4 X
/ acterize Jacobians or hyperelliptic Jacobians. The locus N of abelian varieties
h g,0
t withasingulartheta divisorhascodimension1in g andinabeautiful paper (see
a A
[27]) Mumford calculated its class. But in general not much is known about these
m
Andreotti-Mayer loci N . In particular, we do not even know their codimension.
g,k
:
v In this paper we give estimates for the codimension of these loci. These estimates
i areingeneralnotsharp,butwethink thatthe followingconjecturegivesthe sharp
X
bound.
r
a
Conjecture 1.1. If 1 k g 3 and if N is an irreducible component of N
g,k
whose general point cor≤respo≤nds−to an abelian variety with endomorphism ring Z
then codim (N) k+2 . Moreover, equality holds if and only if one of the
Ag ≥ 2
following happens: (cid:0) (cid:1)
(i) g =k+3 and N = ;
g
H
(ii) g =k+4 and N = .
g
J
We give some evidence for this conjecture by proving the case k = 1. In our
approachweneedtostudythebehaviouroftheAndreotti-Mayerlociatthebound-
ary of the compactified moduli space. A principally polarized (g 1)-dimensional
−
abelian variety (B,Ξ) parametrizes semi-abelian varieties that are extensions of B
1991 Mathematics Subject Classification. 14K10.
1
2 CIROCILIBERTOANDGERARDVANDERGEER
by the multiplicative groupG . This means that B occurs in the boundary of the
m
compactified moduli space ˜ and we can intersect B with the Andreotti-Mayer
g
A
loci. This motivates the definition of loci N (B,Ξ) B for a principally polarized
k
⊂
(g 1)-dimensional abelian variety (B,Ξ). They are formed by the points b in
−
B such that Ξ and its translate Ξ are tangentially degenerate along a subvariety
b
of dimension k. These intrinsically defined subvarieties of an abelian variety are
interestingintheirownrightanddeservefurtherstudy. Theconjectureabovethen
leads to a boundary version that gives a new conjectural answer to the Schottky
problem for simple abelian varieties.
Conjecture1.2. Letk Z . Supposethat(B,Ξ)isasimpleprincipallypolarized
≥1
∈
abelian variety of dimension g not contained in N for all i k. Then there is an
g,i
≥
irreducible component Z of N (B,Ξ) with codim (Z)=k+1 if and only if one of
k B
the following happens:
(i) either g 2, k =g 2 and B is a hyperelliptic Jacobian,
≥ −
(ii) or g 3, k =g 3 and B is a Jacobian.
≥ −
Inourapproachwe willusea specialcompactification ˜ of (see[29,28,5]).
g g
A A
Thepointsoftheboundary∂ ˜ = ˜ correspondtosuitablecompactifications
g g g
A A −A
of g–dimensional semi-abelian varieties. We prove Conjecture 1.1 for k = 1 by
intersecting with the boundary. For higher values of k, the intersection with the
boundary looks very complicated.
2. The universal theta divisor
Letπ : betheuniversalprincipallypolarizedabelianvarietyofrelative
g g
X →A
dimension g over the moduli space of principally polarized abelian varieties of
g
dimensiong overC. In this paper wAe willworkwith orbifoldsandwe shallidentify
(resp. )withthe orbifoldSp(2g,Z)⋉Z2g H Cg (resp.withSp(2g,Z) H ),
g g g g
X A \ × \
where
H = (τ ) Mat(g g,C):τ =τt,Im(τ)>0
g ij
{ ∈ × }
is the usual Siegel upper–half space of degree g. The τ with 1 i j g are
ij
coordinates on H and we let z ,...,z be coordinates on Cg. ≤ ≤ ≤
g 1 g
The Riemann theta function ϑ(τ,z), given on H Cg by
g
×
ϑ(τ,z)= eπi[mtτm+2mtz],
mX∈Zg
isaholomorphicfunctionanditszerolocusisaneffectivedivisorΘ˜ onH Cg which
g
×
descends to a divisor Θ on . If the abelian variety X is a fibre of π, then we let
g
X
Θ be the restrictionofΘtoX. Notethatsinceθ(τ,z)satisfiesθ(τ, z)=θ(τ,z),
X
−
the divisor Θ is symmetric, i.e., ι∗(Θ ) = Θ , where ι = 1 : X X is
X X X X
− →
multiplicationby 1onX. ThedivisorΘ definesthelinebundle (Θ ),which
X X X
− O
yields the principal polarization on X. The isomorphism class of the pair (X,Θ )
X
represents a point ζ of and we will write ζ = (X,Θ ). Similarly, it will be
g X
A
convenienttoidentifyapointξ ofΘwiththeisomorphismclassofarepresentative
triple (X,Θ ,x), where ζ =(X,Θ ) represents π(ξ) and x Θ .
X X g X
∈A ∈
The tangent space to at a point ξ, with π(ξ)=ζ, will be identified with the
g
X
tangent space T T = T Sym2(T ). If ξ = (X,Θ ,x) corresponds
X,x ⊕ Ag,ζ ∼ X,0 ⊕ X,0 X
to the Sp(2g,Z)⋉Z2g–orbit of a point (τ ,z ) H Cg, then the tangent space
0 0 g
T to at ξ can be identified with the tan∈gent×space to H Cg at (τ ,z ),
Xg,ξ Xg g × 0 0
ANDREOTTI–MAYER LOCI AND THE SCHOTTKY PROBLEM 3
which in turn is naturally isomorphic to Cg(g+1)/2+g, with coordinates (a ,b ) for
ij ℓ
1 i,j g and 1 ℓ g that satisfy a = a . We thus view the a ’s as
ij ji ij
co≤ordinat≤es on the ta≤ngen≤t space to H at τ and the b ’s as coordinates on the
g 0 l
tangent space to X or its universal cover.
An important remark is that by identifying the tangent space to at ζ =
g
A
(X,Θ ) withSym2(T ), we canview the projectivizedtangentspaceP(T )=
X X,0 Ag,ζ ∼
P(Sym2(T )) as the linear system of all dual quadrics in Pg−1 = P(T ). In
X,0 X,0
particular,thematrix(a )canbeinterpretedasthematrixdefiningadualquadric
ij
inthe space Pg−1 with homogeneouscoordinates(b :...:b ). Quite naturally,we
1 g
will often use (z :...:z ) for the homogeneous coordinates in Pg−1.
1 g
Recall that the Riemann theta function ϑ satisfies the heat equations
∂ ∂ ∂
ϑ=2π√ 1(1+δ ) ϑ
ij
∂z ∂z − ∂τ
i j ij
for1 i,j g,whereδ istheKroneckerdelta. Weshallabbreviatethisequation
ij
≤ ≤
as
∂ ∂ ϑ=2π√ 1(1+δ )∂ ϑ,
i j − ij τij
where ∂ means the partial derivative∂/∂z and ∂ the partial derivative ∂/∂τ .
j j τij ij
One easily checks that also all derivatives of θ verify the heat equations. We refer
to[38]foranalgebraicinterpretationoftheheatequationsintermsofdeformation
theory.
Ifξ =(X,Θ ,x) ΘcorrespondstotheSp(2g,Z)⋉Z2g–orbitofapoint(τ ,z ),
X 0 0
thentheZariskitang∈entspaceT toΘatξ isthesubspaceofT Cg(g+1)/2+g
Θ,ξ Xg,ξ ≃
defined, with the above conventions, by the linear equation
1
(1) a ∂ ∂ ϑ(τ ,z )+ b ∂ ϑ(τ ,z )=0
ij i j 0 0 ℓ ℓ 0 0
2π√ 1(1+δ )
1≤Xi≤j≤g − ij 1≤Xℓ≤g
in the variables (a ,b ), 1 i,j g, 1 ℓ g. As an immediate consequence we
ij ℓ
≤ ≤ ≤ ≤
get the result (see [35], Lemma (1.2)):
Lemma 2.1. The point ξ =(X,Θ ,x) is a singular point of Θ if and only if x is
X
a point of multiplicity at least 3 for Θ .
X
3. The locus S
g
We begin by defining a suborbifold of Θ supported on the set of points where
π fails to be of maximal rank.
|Θ
Definition 3.1. The closed suborbifold S of Θ is defined on the universal cover
g
H Cg by the g+1 equations
g
×
(2) ϑ(τ,z)=0, ∂ ϑ(τ,z)=0, j =1,...,g.
j
Lemma 2.1implies that the supportof S is the unionof Sing(Θ) andofthe set
g
of smooth points of Θ where π fails to be of maximal rank. Set-theoretically one
|Θ
has
S = (X,Θ ,x) Θ:x Sing(Θ )
g X X
{ ∈ ∈ }
and codim (S ) g+1. It turns out that every irreducible component of S has
Xg g ≤ g
codimension g+1 in (see [8] and an unpublished preprint by Debarre [9]). We
g
X
will come back to this later in 7 and 8.
§ §
4 CIROCILIBERTOANDGERARDVANDERGEER
With the above identification, the Zariski tangent space to S at a given point
g
(X,Θ ,x)of ,correspondingtotheSp(2g,Z)-orbitofapoint(τ ,z ) H Cg,
X g 0 0 g
X ∈ ×
is given by the g+1 equations
a ∂ ϑ(τ ,z )=0,
ij τij 0 0
1≤Xi≤j≤g
(3)
a ∂ ∂ ϑ(τ ,z )+ b ∂ ∂ ϑ(τ ,z )=0, 1 k g
ij τij k 0 0 ℓ ℓ k 0 0 ≤ ≤
1≤Xi≤j≤g 1≤Xℓ≤g
in the variables (a ,b ) with 1 i,j,ℓ g. We will use the following notation:
ij ℓ
≤ ≤
(1) q is the row vector of length g(g+1)/2, given by (∂ θ(τ ,z )), with lexi-
τij 0 0
cographically ordered entries;
(2) q is the row vector of length g(g+1)/2, given by (∂ ∂ θ(τ ,z )), with
k τij k 0 0
lexicographically ordered entries;
(3) M is the g g–matrix (∂ ∂ ϑ(τ ,z )) .
i j 0 0 1≤i,j≤g
×
Then we can rewrite the equations (3) as
(4) a qt =0, a qt +b Mt =0, (j =1,...,g),
· · j · j
where a is the vector (a ) of length g(g + 1)/2, with lexicographically ordered
ij
entries, b is a vector in Cg and M the j–th row of the matrix M.
j
In this setting, the equation (1) for the tangent space to T can be written as:
Θ,ξ
(5) a qt+b ∂ϑ(τ ,z )t =0
0 0
· ·
where ∂ denotes, as usual, the gradient.
Supposenowthepointξ =(X,Θ ,x)inS ,correspondingto(τ ,z ) H Cg
X g 0 0 g
∈ ×
isnotapointofSing(Θ). ByLemma2.1thematrixM isnotzeroandthereforewe
can associate to ξ a quadric Q in the projective space P(T ) P(T ) Pg−1,
ξ X,x X,0
≃ ≃
namely the one defined by the equation
b M bt =0.
· ·
Recallthatb=(b ,...,b )isacoordinatevectoronT andtherefore(b :...:b )
1 g X,0 1 g
are homogeneous coordinates on P(T ). We will say that Q is indeterminate, if
X,0 ξ
ξ Sing(Θ).
∈
The vector q naturally lives in Sym2(T )∨ and therefore, if q is not zero, the
X,0
point [q] P(Sym2(T )∨) determines a quadric in Pg−1 = P(T ). The heat
X,0 X,0
∈
equations imply that this quadric coincides with Q .
ξ
Consider the matrix defining the Zariski tangent space to S at a point ξ =
g
(X,Θ ,x). Wedenotebyr :=r thecorankofthequadricQ ,withtheconvention
X ξ ξ
that r =g if ξ Sing(Θ), i.e., if Q is indeterminate. If we choose coordinates on
ξ ξ
Cg such that the∈first r basis vectors generatethe kernelof q then the shape of the
matrix A of the system (3) is
q 0
g
q1 0g
.
(6) A= . ,
 . 
 
q 0 
 r g
 B
∗ 
ANDREOTTI–MAYER LOCI AND THE SCHOTTKY PROBLEM 5
where q and q areas aboveand B is a (g r) g–matrix with the first r columns
k
− ×
equal to zero and the remaining (g r) (g r) matrix symmetric of maximal
− × −
rank.
Next, we characterize the smooth points ξ = (X,Θ ,x) of S . Before stating
X g
the result, we need one more piece of notation. Given a non-zero vector b =
(b ,...,b ) T , we set ∂ = g b ∂ . Define the matrix ∂ M as the g g–
m1atrix (∂gi∂j∈∂bϑX(,τ00,z0))1≤i,j≤bg. PThℓe=n1 dℓefiℓne the quadric ∂bQξ =b Qξ,b of P(T×X,0)
by the equation
z ∂ M zt =0.
b
· ·
If z =e is the i–th vector of the standard basis, one writes ∂ Q =Q instead of
i i ξ ξ,i
Q for i = 1,...,g. We will use similar notation for higher order derivatives or
ξ,ei
even for differential operators applied to a quadric.
Definition 3.2. We let be the linearsystemofquadricsinP(T )spannedby
ξ X,0
Q
Q and by all quadrics Q with b ker(Q ).
ξ ξ,b ξ
∈
SinceQ hascorankr,thesystem isspannedbyr+1elementsandtherefore
ξ ξ
Q
dim( ) r. This systemmay happentobe empty,but thenQ isindeterminate,
ξ ξ
Q ≤
i.e.,ξliesinSing(Θ). Sometimeswewillusethelowersuffixxinsteadofξ todenote
quadrics and linear systems, e.g. we will sometimes write Q instead of Q , etc.
x ξ
By the heat equations, the linear system is the image of the vector subspace of
ξ
Q
Sym2(T )∨ spanned by the vectors q,q ,...,q .
X,0 1 r
Proposition 3.3. The subscheme S is smooth of codimension g+1 in at the
g g
X
point ξ =(X,Θ ,x) of S if and only if the following conditions are verified:
X g
(i) ξ / Sing(Θ), i.e., Q is not indeterminate and of corank r<g;
ξ
∈
(ii) the linear system has maximal dimension r; in particular, if b ,...,b
ξ 1 r
Q
span the kernel of Q , then the r + 1 quadrics Q , Q ,...,Q are
ξ ξ ξ,b1 ξ,br
linearly independent.
Proof. The subscheme S is smooth of codimension g+1 in at ξ if and only if
g g
X
the matrix A appearing in (6) has maximal rank g+1. Since the submatrix B of
A has rank g r, the assertion follows. (cid:3)
−
Corollary 3.4. If Q is a smooth quadric, then S is smooth at ξ =(X,Θ ,x).
ξ g X
4. Quadrics and Cornormal Spaces
Next we study the differential of the restriction to S of the map π :
g g g
X → A
at a point ξ = (X,Θ ,x) S . We are interested in the kernel and the image of
X g
dπ . We can view these∈spaces in terms of the geometry of Pg−1 = P(T ) as
|Sg,ξ X,0
follows:
Π =P(ker(dπ )) P(T )
ξ |Sg,ξ ⊆ X,0
is a linear subspace of P(T ) and
X,0
Σ =P(Im(dπ )⊥) P(Sym2(T )∨)
ξ |Sg,ξ ⊆ X,0
is a linear system of quadrics in P(T ).
X,0
The following proposition is the key to our approach; we use it to view the
quadrics as elements of the conormal space to our loci in the moduli space.
Proposition 4.1. Let ξ =(X,Θ ,x) be a point of S . Then:
X g
6 CIROCILIBERTOANDGERARDVANDERGEER
(i) Π is the vertex of the quadric Q . In particular, if ξ is a singular point of
ξ ξ
Θ, then Π is the whole space P(T );
ξ X,0
(ii) Σ contains the linear system .
ξ ξ
Q
Proof. The assertions follow from the shape of the matrix A in (6). (cid:3)
Thispropositiontellsusthat,givenapointξ =(X,Θ ,x) S ,themapdπ
X ∈ g |Sg,ξ
is not injective if and only if the quadric Q is singular.
ξ
The orbifold S is stratified by the corank of the matrix (∂ ∂ θ).
g i j
Definition 4.2. For 0 k g we define S as the closed suborbifold of S
g,k g
defined by the equations≤on H≤ Cg
g
×
ϑ(τ,z)=0, ∂ ϑ(τ,z)=0, (j =1,...,g),
j
(7)
rk (∂ ∂ ϑ(τ,z)) g k.
i j 1≤i,j≤g
≤ −
(cid:0) (cid:1)
Geometrically this means that ξ S if and only if dim(Π ) k 1 or
g,k ξ
∈ ≥ −
equivalently Q has corank at least k. We have the inclusions
ξ
S =S S ... S =S Sing(Θ)
g g,0 g,1 g,g g
⊇ ⊇ ⊇ ∩
andS isthelocuswherethemapdπ isnotinjective. ThelociS havebeen
g,1 |Sg,ξ g,k
considered also in [16].
We have the following dimension estimate for the S .
g,k
Proposition 4.3. Let 1 k g 1 and let Z be an irreducible component of S
g,k
≤ ≤ −
not contained in S . Then we have
g,k+1
k+1
codim (Z) .
Sg ≤(cid:18) 2 (cid:19)
Proof. Locally, in a neighborhood U in S of a point z of Z S we have a
g g,k+1
morphism f : U , where is the linear system of all quad\rics in Pg−1. The
→ Q Q
map f sends ξ = (X,Θ ,x) U to Q . The scheme S is the pull–back of the
X ξ g,k
∈
subscheme of formedbyallquadricsofcorankk. Sincecodim ( )= k+1 ,
the assertioQnkfolloQws. Q Qk (cid:0) 2 (cid:3)(cid:1)
Using the equations (7) it is possible to make a local analysis of the schemes
S ,e.g. itispossibletowritedownequationsfortheirZariskitangentspaces(see
g,k
6 for the case k = g). This is however not particularly illuminating, and we will
§
not dwell on this here.
Itis usefultogiveaninterpretationofthe pointsξ =(X,Θ ,x) S interms
X g,k
∈
of singularities of the theta divisor Θ . Suppose that ξ is such that Sing(Θ )
X X
containsasubschemeisomorphictoSpec(C[ǫ]/(ǫ2))supportedatx. Thissubscheme
of X is given by a homomorphism
C[ǫ]/(ǫ2), f f(x)+∆(1)f(x) ǫ,
X,x
O → 7→ ·
where ∆(1) is a non–zero differential operator of order 1, hence ∆(1) = ∂ , for
b
some non–zero vector b Cg. Then the condition Spec≤(C[ǫ]/(ǫ2)) Sing(Θ ) is
X
∈ ⊂
equivalent to saying that ϑ and ∂ ϑ satisfy the equations
b
(8) f(τ ,z )=0, ∂ f(τ ,z )=0, 1 j g,
0 0 j 0 0
≤ ≤
and this, in turn, is equivalent to the fact that the quadric Q is singular at the
ξ
point [b].
ANDREOTTI–MAYER LOCI AND THE SCHOTTKY PROBLEM 7
More generally, we have the following proposition, which explains the nature of
the points in S for k <g.
g,k
Proposition 4.4. Suppose that x Sing(Θ ) does not lie on Sing(Θ). Then
X
Sing(Θ ) contains a scheme isomorp∈hic to Spec(C[ǫ ,...,ǫ ]/(ǫ ǫ : 1 i,j k <
X 1 k i j
≤ ≤
g)) supported at x if and only if the quadric Q has corank r k. Moreover, the
ξ
≥
Zariski tangent space to Sing(Θ ) at x is the kernel space of Q .
X ξ
Proof. With a suitable choice of coordinates in X, the condition that the scheme
Spec(C[ǫ ,...,ǫ ]/(ǫ ǫ : 1 i,j k < g)) is contained in Sing(Θ ) is equivalent
1 k i j X
≤ ≤
to the fact that the functions ϑ and ∂ ϑ for i = 1,...,k satisfy (8). But this the
i
same as saying that ∂ ∂ ϑ(τ ,z ) is zero for i = 1,...,k, j = 1,...,g, and the
i j 0 0
vectors e , i=1,...,k, belong to the kernel of Q . This settles the first assertion.
i ξ
The scheme Sing(Θ ) is defined by the equations (2), where τ is now fixed
X
and z is the variable. By differentiating, and using the same notation as above,
we see that the equations for the Zariski tangent space to Sing(Θ ) at x are
X
g b ∂ ∂ ϑ(τ ,z ), j = 1,...,g i.e., b M = 0, which proves the second as-
sPerit=io1n.i i j 0 0 · (cid:3)
5. Curvi-linear subschemes in the singular locus of theta
A 0–dimensionalcurvi-linearsubscheme Spec(C[t]/(tN+1)) X of lengthN+1
⊂
supported at x is given by a homomorphism
N
(9) δ : C[t]/(tN+1), f ∆(j)f(x) tj,
X,x
O → 7→ ·
Xj=0
with ∆(j) a differential operator of order j, j = 1,...,N, with ∆(N) non–zero,
≤
and∆(0)(f)=f(x). Theconditionthatthemapδisahomomorphismisequivalent
to saying that
k
(10) ∆(k)(fg)= ∆(r)f ∆(k−r)g, k =0,...,N
·
Xr=0
for any pair (f,g) of elements of . Two such homomorphisms δ and δ′ define
X,x
O
the same subscheme if andonly if they differ by compositionwitha automorphism
of C[t]/(tN+1).
Lemma5.1. Themapδ definedin (9)isahomomorphism ifandonlyifthereexist
translationinvariantvectorfieldsD ,...,D onX suchthatforeveryk =1,...,N
1 N
one has
1
(11) ∆(k) = Dh1 Dhk.
h ! h ! 1 ··· k
h1+2h2+X...+khk=k>0 1 ··· k
Moreover, two N–tuples of vector fields (D ,...,D ) and (D′,...,D′ ) determine
1 N 1 N
the same 0–dimensional curvi-linear subscheme of X of length N +1 supported at
a given point x X if and only if there are constants c ,...,c , with c =0, such
1 N 1
∈ 6
that
i
D′ = ci−j+1D , i=1,...,N.
i j j
Xj=1
8 CIROCILIBERTOANDGERARDVANDERGEER
Proof. Ifthe differentialoperators∆(k), k =1,...,N,areasin(11),one computes
that (10) holds, hence δ is a homomorphism.
As for the converse, the assertion trivially holds for k = 1. So we proceed by
induction on k. Write ∆(k) = k D(k), where D(k) is the homogeneous part
i=1 i i
of degree i, and write D insteaPd of D(k). Using (10) one verifies that for every
k 1
k =1,...,N and every positive i k one has
≤
k−i+1
iD(k) = D D(k−j).
i j i−1
Xj=1
Formula (11) follows by induction and easy combinatorics.
To prove the final assertion, use the fact that an automorphism of C[t]/(tN+1)
is determined by the image c t+c t2+...+c tN of t, where c =0. (cid:3)
1 2 N 1
6
In formula (11) one has h 1. If ∆(1) = D then ∆(2) = 1D2+D , ∆(3) =
k ≤ 1 2 1 2
(1/3!)D3+(1/2)D D +D etc.
1 1 2 3
Eachnon-zerosummandin(11)isoftheform(1/h ! h !)Dhi1 Dhiℓ,where
i1 ··· iℓ i1 ··· iℓ
1 i < ...< i k, i h +...+i h =k and h ,...,h are positive integers.
≤ 1 ℓ ≤ 1 i1 ℓ iℓ i1 iℓ
Thus formula (11) can be written as
(12) ∆(k) = 1 Dhi1 Dhiℓ,
h ! h ! i1 ··· iℓ
{hi1X,...,hiℓ} i1 ··· iℓ
where the subscript h ,...,h means that the sum is taken over all ℓ–tuples of
{ i1 iℓ}
positiveintegers(h ,...,h )with1 i < <i k andi h + +i h =k.
i1 iℓ ≤ 1 ··· ℓ ≤ 1 i1 ··· ℓ iℓ
Remark 5.2. Letx X correspondtothe pair(τ ,z ). Thedifferentialoperators
0 0
∈
∆(k), k =1,...,N, defined as in (11) or (12) have the following property: if f is a
regular function such that ∆(i)f satisfies (8) for all i = 0,...,k 1, then one has
−
∆(k)f(τ ,z )=0.
0 0
Wewantnowtoexpresstheconditionsinorderthata0–dimensionalcurvi-linear
subschemeofX oflengthN+1supportedatagivenpointx X correspondingto
∈
the pair (τ ,z ) and determined by a given N–tuple of vector fields (D ,...,D )
0 0 1 N
lies in Sing(Θ ). To do so, we keep the notation we introduced above.
X
Let us write D = g η ∂ , so that D corresponds to the vector η =
i ℓ=1 iℓ ℓ i i
(ηi1,...,ηig). As beforePwe denote by M the matrix (∂i∂jθ(τ0,z0)).
Proposition 5.3. The 0–dimensional curvi-linear subscheme R of X of length
N+1,supportedatthepointx X correspondingtothepair(τ ,z )anddetermined
0 0
∈
by the N–tuple of vector fields (D ,...,D ) lies in Sing(Θ ) if and only if x
1 N X
∈
Sing(Θ ) and moreover for each k =1,...,N one has
X
(13) 1 η ∂hi1 ∂hiℓ−1M =0,
{hi1X,...,hiℓ}hi1!···hiℓ! iℓ · ηi1 ··· ηiℓ
where the sum is taken over all ℓ–tuples of positive integers (h ,...,h ) with 1
i1 iℓ ≤
i < <i k and i h + +i h =k.
1 ··· ℓ ≤ 1 i1 ··· ℓ iℓ
Proof. The scheme R is contained in Sing(Θ ) if and only if one has
X
∆(k)θ(τ ,z )=0, ∂ ∆(k)θ(τ ,z )=0 k =0,...,N, j =1,...,g.
0 0 j 0 0
ANDREOTTI–MAYER LOCI AND THE SCHOTTKY PROBLEM 9
By Remark 5.2, this is equivalent to
θ(τ ,z )=0, ∂ ∆(k)θ(τ ,z )=0 k =0,...,N, j =1,...,g.
0 0 j 0 0
The assertion follows by the expression (12) of the operators ∆(k). (cid:3)
Forinstance,considertheschemeR ,supportedatx Sing(Θ ),corresponding
1 X
∈
to the vector field D . Then R is contained in Sing(Θ ) if and only if
1 1 X
(14) η M =0.
1
·
This agrees with Proposition 4.4. If R is the scheme supported at x and corre-
2
sponding to the pair of vector fields (D ,D ), then R is contained in Sing(Θ ) if
1 2 2 X
and only if, besides (14) one has also
(15) (1/2)η ∂ M +η M =0.
1· η1 2·
Next, consider the scheme R supported at x and corresponding to the triple of
3
vectorfields(D ,D ,D ). ThenR iscontainedinSing(Θ )ifandonlyif,besides
1 2 3 3 X
(14) and (15) one has also
(16) (1/3!)η ∂2 M +(1/2)η ∂ M +η M =0
1· η1 2· η1 3·
and so on. Observe that (13) can be written in more than one way. For example
η ∂ M =η ∂ M so that (16) could also be written as
2· η1 1· η2
(1/3!)η ∂2 M +(1/2)η ∂ M +η M =0.
1· η1 1· η2 3·
So far we have been working in a fixed abelian variety X. One can remove
this restriction by working on S and by letting the vector fields D ,...,D vary
g 1 N
with X, which means that we let the vectors η depend on the variables τ . Then
i ij
the equations (13) define a subscheme S (D) of Sing(Θ) which, as a set, is the
g
locus of all points ξ =(X,Θ ,x) S such that Sing(Θ ) contains a curvi–linear
X g X
∈
scheme of length N +1 supported at x, corresponding to the N–tuple of vector
fields D =(D ,...,D ), computed on X.
1 N
One can compute the Zariski tangent space to S (D) at a point ξ =(X,Θ ,x)
g X
in the same way, and with the same notation, as in 3. This gives in general
§
a complicated set of equations. However we indicate one case in which one can
draw substantial information from such a computation. Consider indeed the case
in which D = ... = D = 0, and call b the corresponding tangent vector to X
1 N
6
at the origin, depending on the the variables τ . In this case we use the notation
ij
D = (D ,...,D ) and we denote by R the corresponding curvi–linear
b,N 1 N x,b,N
scheme supported at x. For a given such D = (D ,...,D ), consider the linear
1 N
system of quadrics
Σ (D)=P(Im(dπ )⊥)
ξ |Sg(D),ξ
inP(T ). Onehasagainaninterpretationofthesequadricsintermsofthenormal
X,0
space:
Proposition 5.4. In the above setting, the space Σ (D ) contains the quadrics
ξ b,N
Q ,∂ Q ,...,∂NQ .
ξ b ξ b ξ
Proof. The equations (13) take now the form
θ(τ,z)=0, ∂ θ(τ,z)=0, i=1,...,g
i
b M =b ∂ M = =b ∂N−1M =0.
· · b ··· · b
By differentiating the assertion immediately follows. (cid:3)
10 CIROCILIBERTOANDGERARDVANDERGEER
6. Higher multiplicity points of the theta divisor
We now study the case of higher order singularities on the theta divisor. For
a multi-index I = (i ,...,i ) with i ,...,i non-negative integers we set zI =
1 g 1 g
zi1 zig anddenoteby∂ the operator∂i1 ∂ig. Moreover,welet I = g i ,
1 ··· g I 1 ··· g | | ℓ=1 ℓ
which is the length of I and equals the order of the operator ∂I. P
Definition 6.1. For a positive integer r we let S(r) be the subscheme of which
g g
is defined on H Cg by the equations X
g
×
(17) ∂ ϑ(τ,z)=0, I =0,...,r 1.
I
| | −
One has the chain of subschemes
... S(r) ... S(3) S(2) =S S(1) =Θ
⊆ g ⊆ ⊆ g ⊆ g g ⊂ g
andasasetS(r) = (X,Θ ,x) Θ:x has multiplicity r for Θ . Onedenotes
g X X
{ ∈ ≥ }
by Sing(r)(Θ ) the subscheme of Sing(Θ ) formed by all points of multiplicity at
X X
least r. One knows that S(r) = as soon as r >g (see [36]). We can compute the
g
∅
Zariski tangent space to S(r) at a point ξ =(X,Θ ,x) in the same vein, and with
g X
thesamenotation,asin 3. Takingintoaccountthatθ andallitsderivativesverify
§
the heat equations, we find the equations by replacing in (3) the term θ(τ ,z ) by
0 0
∂ θ(τ ,z ).
I 0 0
As in 3,wewishtogivesomegeometricalinterpretation. Forinstance,wehave
§
the following lemma which partially extends Lemma 2.1 or 3.3.
Lemma 6.2. For every positive integer r the scheme S(r+2) is contained in the
g
singular locus of S(r).
g
Nextweareinterestedinthedifferentialoftherestrictionofthemapπ :
g g
X →A
toS(r) atapointξ =(X,Θ ,x)whichdoesnotbelongtoS(r+1). Thismeansthat
g X g
Θ has a point of multiplicity exactly r at x. If we assume, as we may, that x is
X
the origin of X , i.e. z =0, then the Taylor expansion of θ has the form
0
∞
ϑ= ϑ ,
i
Xi=r
where ϑ is a homogeneous polynomial of degree i in the variables z ,...,z and
i 1 g
1
θ = ∂ θ(τ ,z )zI
r I 0 0
i ! i !
I=(i1,.X..,ig),|I|=r 1 ··· g
is not identically zero. The equation θ =0 defines a hypersurface TC of degree r
r ξ
in Pg−1 =P(T ), which is the tangent cone to Θ at x.
X,0 X
We willdenotebyVert(TC )thevertexofTC ,i.e.,the subspaceofPg−1 which
ξ ξ
is the locus of points of multiplicity r of TC . Note that it may be empty. In case
ξ
r = 2, the tangent cone TC is the quadric Q introduced in 3 and Vert(TC ) is
ξ ξ ξ
§
its vertex Π .
ξ
More generally, for every s r, one can define the subscheme TC(s) =TC(s) of
≥ ξ x
Pg−1 =P(T ) defined by the equations
X,0
θ =...=θ =0,
r s
which is called the asymptotic cone of order s to Θ at x.
X