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 degenerate 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 r0 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 < 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

Andreotti-Mayer loci and the Schottky problem

Similar Andreotti-Mayer loci and the Schottky problem