Semicontinuity of Gauss maps and the Schottky problem
aa r X i v : . [ m a t h . AG ] D ec SEMICONTINUITY OF GAUSS MAPSAND THE SCHOTTKY PROBLEM
GIULIO CODOGNI AND THOMAS KR ¨AMER
Abstract.
We show that the degree of Gauss maps on abelian varieties issemicontinuous in families, and we study its jump loci. As an applicationwe obtain that in the case of theta divisors this degree answers the Schottkyproblem. Our proof computes the degree of Gauss maps by specialization ofLagrangian cycles on the cotangent bundle. We also get similar results forthe intersection cohomology of varieties with a finite morphism to an abelianvariety; it follows that many components of Andreotti-Mayer loci, includingthe Schottky locus, are part of the stratification of the moduli space of ppav’sdefined by the topological type of the theta divisor.
Contents
1. Introduction 12. Lagrangian specialization 63. Jump loci for the degree 84. Generalities about families of rational maps 125. Gauss maps on abelian varieties 136. Application to the Schottky problem 147. A topological view on jump loci 15References 191.
Introduction
The Gauss map of a hypersurface in projective space is the rational map thatsends any smooth point of the hypersurface to its normal direction in the dualprojective space. The analogous notion of Gauss map for subvarieties of abelianvarieties appears already in Andreotti’s proof of the Torelli theorem [2]. In contrastto the case of projective hypersurfaces, the Gauss map for any ample divisor on anabelian variety is generically finite of degree >
1, and its degree is related to thesingularities of the divisor. We show that this degree is lower semicontinuous infamilies, and we study its jump loci. As an application we get that in the modulispace of principally polarized abelian varieties, the degree of the Gauss map refinesthe Andreotti-Mayer stratification and answers the Schottky problem as conjecturedin [11]. We also obtain similar results for the intersection cohomology of varietieswith a finite morphism to an abelian variety. In particular, many Andreotti-Mayerloci such as the Schottky locus are determined by the topological type of the thetadivisor. We work over an algebraically closed field k with char( k ) = 0. Mathematics Subject Classification.
Primary 14K12; Secondary 14C17, 14F10, 14H42.
Key words and phrases.
Gauss map, abelian variety, theta divisor, Schottky problem.
GIULIO CODOGNI AND THOMAS KR¨AMER
Let A be an abelian variety over k . Bytranslations we may identify its tangent spaces at all points, hence the cotangentbundle T ∨ A = A × V is trivial with fiber V = H ( A, Ω A ). The Gauss map of areduced effective divisor D ⊂ A is the rational map γ D : D P V that sends a smooth point of the divisor to its conormal direction at that point;it coincides with the rational map given by the linear series P V ∨ = | O D ( D ) | . Foran irreducible divisor this is a generically finite dominant map iff the divisor isample, which happens iff the divisor is not stable under translations by any positivedimensional abelian subvariety [31, cor. II.11, lem. II.9]. Even in the genericallyfinite case the Gauss map can have positive dimensional fibers [4].For algebraic families of generically finite maps the generic degree always definesa constructible stratification of the parameter space, but in general it can jumpin both directions (see example 4.2). Our first semicontinuity result says that forGauss maps on abelian varieties this does not happen: Theorem 1.1.
Let A → S be an abelian scheme over a variety S , and let D ⊂ A be a relatively ample divisor which is flat over S . Let D s ⊂ A s denote their fibersover s ∈ S , and let γ D s be the corresponding Gauss map. Then for each d ∈ N thesubsets S d = (cid:8) s ∈ S | deg( γ D s ) ≤ d (cid:9) ⊆ S are closed in the Zariski topology. The above result does not show where the degree actually jumps. Let us saythat an irreducible subvariety of an abelian variety is negligible if it is stable undertranslations by a positive dimensional abelian subvariety. Simple abelian varietieshave no negligible subvarieties other than themselves. More generally, by [1, th. 3]an irreducible closed subvariety of an abelian variety is negligible iff it is not ofgeneral type. Our second result says that in the setting of theorem 1.1 the degreeof the Gauss map jumps whenever a new non-negligible component appears in thesingular locus Sing( D s ). To make this precise we specify a curve along which wemove inside the parameter space: Theorem 1.2.
Let S ′ ⊂ S be a curve, and fix a point ∈ S ′ ( k ) . If Sing( D ) has an irreducible component which is not negligible and not in the Zariski closureof S s ∈ S ′ \{ } Sing( D s ) , then deg( γ D ) < deg( γ D s ) for all s ∈ S ′ ( k ) \ { } in some Zariski open neighborhood of the point . The above in particular applies if the singular locus has no negligible componentsand satisfies dim(Sing( D )) > dim(Sing( D s )) for all s = 0. This last condition ismotivated by the case of theta divisors and the Schottky problem. Let A g be the moduli space ofprincipally polarized abelian varieties of dimension g . Inside it, consider for d ∈ N the Gauss loci G d = { ( A, Θ) ∈ A g | deg( γ Θ ) ≤ d } as in [11, sect. 4] The above results show that these loci are closed (cor. 6.1) andrefine the Andreotti-Mayer stratification (cor. 6.3). Thus the Gauss loci provide EMICONTINUITY OF GAUSS MAPS 3 a solution for the Schottky problem to characterize the closure of the locus ofJacobians in the moduli space of principally polarized abelian varieties:
Corollary 1.3.
Inside A g we have:(a) The locus of Jacobians is a component of G d for d = (cid:0) g − g − (cid:1) .(b) The locus of hyperelliptic Jacobians is a is a component of G d for d = 2 g − . The above corollary is shown in section 6 together with an analogous statementfor Prym varieties. It confirms a conjecture by the first author, Grushevsky andSernesi [11, conjecture 1.6] who verified it for g ≤ D -modules allows to refine the degree of the Gauss map to representation theoreticinvariants that might distinguish the Jacobian locus [25]. For the proof of theorem 1.1 and 1.2we interpret the degree of the Gauss map as an intersection number of Lagrangiancycles on the cotangent bundle of the abelian variety and apply specialization forsuch cycles [18, 33], which we can do because char( k ) = 0. To explain how thisworks, let us forget about abelian varieties for a moment and fix any ambientsmooth variety W over k . The conormal variety to a subvariety X ⊂ W is definedas the closureΛ X = { ( x, ξ ) | x ∈ Sm( X ) , ξ ∈ T ∨ x ( W ) , ξ ⊥ T x ( X ) } ⊂ T ∨ ( W )of the conormal bundle to the smooth locus Sm( X ), where the closure is takenin the total space of the cotangent bundle of the ambient smooth variety. Thisconormal variety always has pure dimension n = dim( W ), in fact it is Lagrangianwith respect to the natural symplectic structure on the cotangent bundle. It is alsoconic, i.e. stable under the natural action of the multiplicative group on the fibersof the cotangent bundle. Conversely, any closed conic Lagrangian subvariety of thecotangent bundle arises like this [22, lemma 3]. So the map X Λ X induces anisomorphism Z ( W ) = { cycles on W } ∼ −→ L ( W ) = { conic Lagrangian cycles on T ∨ W } , where by a conic Lagrangian cycle we mean a Z -linear combination of closed conicLagrangian subvarieties. In the case of projective varieties we can talk about thedegree of conormal varieties: Definition 1.4. If W is projective, the degree homomorphism on conic Lagrangiancycles is the mapdeg : L ( W ) −→ CH n ( T ∨ ( W )) i ∗ −→ CH n ( W ) ։ Z which is given by the intersection number with the zero section i : W ֒ → T ∨ ( W ). Example 1.5.
Over the complex numbers the above degree can be computed asfollows. For any constructible function F : W → Z consider the topological Eulercharacteristic χ top ( W, F ) = X n ∈ Z n · χ top ( F − ( n )) , GIULIO CODOGNI AND THOMAS KR¨AMER where χ top ( F − ( n )) denotes the alternating sum of the Betti numbers of F − ( n ). Inthe theory of characteristic classes of singular varieties this definition is applied toa particular constructible function Eu X : W → Z , the local Euler obstruction of asubvariety X ⊆ W [28, sect. 3]. Outside of the singular locus Sing( X ) ⊆ X it hasthe form Eu X ( p ) = ( p ∈ W \ X, p ∈ Sm( X ) , but its values on the singular locus depend on the singularities: For instance, if X is a curve, Eu X ( p ) is the multiplicity of that curve at p . In general, the degree ofconormal varieties in definition 1.4 can be expressed as an Euler characteristic bythe formula deg(Λ X ) = ( − dim( X ) · χ top ( X, Eu X ) , see [33, lemme 1.2.1] [16, prop. 6.1(b)]. The right hand side can be computed easilyas soon as we know the local Euler obstruction. For a smooth rational curve X we get deg(Λ X ) = − X ) = −
3. Notethat a cuspidal cubic is homeomorphic to a smooth rational curve, so the degree ofconormal varieties is not a topological invariant. Moreover, it can be negative.
Of course there are no rationalcurves in abelian varieties, and in the case of abelian varieties the degree behavesmuch better. By [37, th. 1] we have the following result (see section 5):
Proposition 1.6. If W = A is an abelian variety, then • deg(Λ X ) ≥ for any X ⊂ A , • deg(Λ X ) = 0 if and only if X is negligible. • deg(Λ X ) = deg( γ X ) for divisors X ⊂ A with Gauss map γ X . This easily implies theorem 1.1 when combined with the principle of Lagrangianspecialization which we recall in section 2: For any flat family of subvarieties ina smooth ambient 1-parameter family, the limit of their conormal varieties is aneffective conic Lagrangian cycle whose support contains the conormal variety tothe central fiber as a component, and the total degree of the limit cycle equals thedegree of a general fiber. The same argument shows that our semicontinuity resultholds not only for divisors but for subvarieties of any codimension:
Theorem 1.7.
Let A → S be an abelian scheme over a variety S , and let X ⊂ A be an arbitrary family of subvarieties which is flat over S . Then for each d ∈ N thesubsets S d = (cid:8) s ∈ S | deg(Λ X s ) ≤ d (cid:9) ⊆ S are closed in the Zariski topology. It remains to prove theorem 1.2. Given the interpretation for the degree of Gaussmaps in prop. 1.6, the proof has nothing to do with abelian varieties: In section 3we show that for any flat family of divisors on a smooth 1-parameter variety, thespecialization of their conormal varieties contains an extra component wheneverthe singular locus of the fiber jumps. While the final criterion is phrased only fordivisors, we formulate our arguments as far as possible for subvarieties in arbitrarycodimension to get beyond theorem 1.2 (see example 3.6). This is important evenif one only wants to study singularities of divisors: In the theory of Chern classes
EMICONTINUITY OF GAUSS MAPS 5 for singular varieties one attaches to any subvariety X ⊂ A a characteristic cycleof the form Λ = Λ X + X Z m Z Λ Z where Z runs through certain strata in Sing( X ) [22, 33], and the topologicallymeaningful invariant that appears in generalizations of the Gauss-Bonnet indexformula is the total degree deg(Λ) involving all the strata. In section 7, which is not used in the restof the paper, we deduce from our previous results a general semicontinuity theoremfor the intersection cohomology of varieties over the complex numbers. Recall thatfor a complex variety X , the intersection cohomology IH • ( X ) only depends on itshomeomorphism type in the Euclidean topology; it coincides with Betti cohomologyin the smooth case but is better behaved in general [6, 19, 20, 23, 29]. We denoteby χ IC ( X ) = X i ≥ ( − i +dim( X ) dim IH i ( X )the Euler characteristic of the intersection cohomology. This Euler characteristicis usually not semicontinuous in families, it can jump in both directions. But forfamilies of finite branched covers of subvarieties in complex abelian varieties thisdoes not happen (see lemma 7.6 and corollary 7.7): Theorem 1.8.
Let f : X → S be a family of varieties such that each fiber X s isgenerically reduced and admits a finite morphism to an abelian variety. Then foreach d ∈ N the loci S d := { s ∈ S | χ IC ( X s ) ≤ d } ⊆ S are closed in the Zariski topology. This puts our results in a topological context, since the intersection cohomologyof a complex variety only depends on its homeomorphism type. For instance, itfollows from the above that a singular theta divisor cannot be homeomorphic to asmooth one (recall that there are examples of normal varieties which are singularbut homemorphic to smooth varieties, such as those by Brieskorn [9, 10]). Incorollary 7.9 we will see that the Jacobian locus appears in the stratification of A g by the intersection cohomology of the theta divisor, so we obtain: Corollary 1.9.
The locus of Jacobian varieties in A g is an irreducible componentof the closure of the locus of all ppav’s whose theta divisor is homeomorphic to atheta divisor on a Jacobian variety. It seems an interesting problem to study the topology of theta divisors on abelianvarieties in more detail.
Acknowledgements.
The authors would like to thank Sam Grushevsky, AriyanJavanpeykar, Constantin Podelski, Claude Sabbah and Edoardo Sernesi for helpfulcomments and discussions. G.C. is funded by the MIUR
Excellence DepartmentProject , awarded to the Department of Mathematics, University of Rome, Tor Ver-gata, CUP E83C18000100006, and PRIN 2017
Advances in Moduli Theory andBirational Classification . T.K. is supported by DFG Research Grant 430165651
Characteristic Cycles and Representation Theory . GIULIO CODOGNI AND THOMAS KR¨AMER Lagrangian specialization
For convenience we include in this section a self-contained review of some basicfacts about the specialization of Lagrangian cycles, which was introduced in relationwith Chern-MacPherson classes [33] and nearby cycles for D -modules and perversesheaves [18]. We work in a relative setting over a smooth curve S . The family of ourambient spaces is given by a smooth dominant morphism of varieties f : W → S where dim( W ) = n + 1. Let X ⊂ W be a reduced closed subvariety. The relativesmooth locusSm( X/S ) = { x ∈ Sm( X ) | the restriction f | X : X → S is smooth at x } is nonempty iff dim f ( X ) >
0, in which case X → S is flat and Sm( X/S ) ⊂ X isan open dense subset because char( k ) = 0. Any x ∈ Sm(
X/S ) is a smooth point ofthe fiber X s = f − ( s ) ∩ X over the image point s = f ( x ). Hence inside the totalspace T ∨ ( W/S ) = T ∨ ( W ) /f − T ∨ ( S )of the relative cotangent bundle, we define the relative conormal variety to X asthe closureΛ X/S = { ( x, ξ ) ∈ T ∨ ( W/S ) | x ∈ Sm(
X/S ) , ξ ⊥ T x X f ( x ) } ⊂ T ∨ ( W/S ) . Remark 2.1. In [8] the relative conormal variety is instead defined as the closureinside the absolute cotangent bundle. This notion of relative conormal variety isobtained from ours by base change via the quotient map T ∨ ( W ) ։ T ∨ ( W/S ) , i.e. wehave T ∨ ( W ) × T ∨ ( W/S ) Λ X/S = { ( x, ξ ) ∈ T ∨ ( W ) | x ∈ Sm(
X/S ) , ξ ⊥ T x X f ( x ) } . Proof.
Both sides are irreducible closed subvarieties of T ∨ ( W ) | X . Indeed, for theright hand side this holds by definition, for the left hand side it follows from the factthat Λ X/S ⊂ T ∨ ( X/S ) is an irreducible closed subvariety and T ∨ W → T ∨ ( X/S ) isa fibration with irreducible fibers. So it suffices to show that both sides agree oversome open dense U ⊂ X . We can assume X is flat over S and take U = Sm( X/S ),in which case the result is obvious. (cid:3)
Lemma 2.2. If X is flat and irreducible over S , then so is Λ X/S .Proof. Λ X/S is defined as the schematic closure of a locally closed subscheme V ofthe relative cotangent bundle T ∨ ( A/S ). The subscheme V is the total space of avector bundle over a smooth variety, so it is a smooth variety as well. Its schematicclosure is integral, and a morphism from an integral scheme to a smooth curve isflat iff it is dominant. (cid:3) Relative conormal varieties can be seen as families of conormal varieties. In whatfollows we denote by L ( W/S ) = M X ⊂ W Z · Λ X/S the free abelian group on relative conormal varieties to closed subvarieties X ⊂ W that are flat over S . By the specialization of Λ ∈ L ( W/S ) at s ∈ S ( k ) we meanthe cycle sp s (Λ) = (cid:2) Λ · f − ( s ) (cid:3) EMICONTINUITY OF GAUSS MAPS 7 which underlies the schematic fiber of the morphism Λ
X/S → S at s . This is againa conic Lagrangian cycle by the following classical result, see [17, prop. (a), p. 179]or in an analytic setup [27, sect. 1.2]: Lemma 2.3 ( Principle of Lagrangian specialization ) . The specialization at s gives a homomorphism sp s : L ( W/S ) −→ L ( W s ) sending effective cycles to effective cycles. On Chow groups it induces the Gysinmap in the bottom row of the following commutative diagram: L ( W/S ) L ( W s )CH n ( T ∨ ( W/S )) CH n − ( T ∨ ( W s )) sp s i ∗ s For any closed subvariety X ⊂ W which is flat over S , there is a finite subset Σ ⊂ S such that sp s (Λ X/S ) = ( Λ X s for s ∈ S \ Σ ,m X s Λ X s + P Z ⊂ Sing( X s ) m Z Λ Z for s ∈ Σ , where m X s , m Z > and the sum runs over finitely many subvarieties Z ⊂ Sing( X s ) .Proof. Note that T ∨ ( W s ) is an effective Cartier divisor in T ∨ ( W ). It intersectsproperly any relative conormal variety to a subvariety which is flat over S . Henceit is clear that the specialization induces on Chow groups the Gysin map definedin [16, sect. 2.6] and sends effective cycles to effective cycles.Now take an irreducible subvariety X ⊂ W which is flat over S . By Lemma 2.2the morphism Λ X/S → S is flat and hence all its fibers are pure dimensional ofthe same dimension. Furthermore the action of the multiplicative group preservesthe fibers of T ∨ ( W/S ) → S and so the fibers of Λ X/S → S are unions of conicsubvarieties. As the canonical relative symplectic form on T ∨ ( W/S ) restricts tothe canonical symplectic form on T ∨ ( W s ) for every s , we conclude that the fibersof Λ X/S → S are also Lagrangian and hence a union of conormal varieties, since theconic Lagrangian subvarieties of the cotangent bundle are precisely the conormalvarieties [22, lemma 3]. The coefficients are non-negative as the specialization ofeffective cycles is effective. Hence sp s (Λ X/S ) is a sum of conormal varieties, andsince under the morphism T ∨ ( A s ) → A s its support surjects onto X s , we concludethat one of the appearing components must be Λ X s .As Λ X/S is irreducible and we work over a field of characteristic zero, there existsa Zariski open dense subset of S over which the fibers of the morphism Λ X/S → S are reduced and irreducible. We conclude that for s in this Zariski open densesubset of S we have sp s (Λ X/S ) = Λ X s . Moreover, the specialization cannot haveany further components over the relative smooth locus Sm( X/S ) ⊆ X , since onthat locus also the morphism Λ X/S → S restricts to a smooth morphism. (cid:3) We have the following consequence of flatness:
GIULIO CODOGNI AND THOMAS KR¨AMER
Proposition 2.4.
Let X ⊂ W be a closed subvariety which is flat over S , then thedegree d = deg(sp s (Λ X/S )) is independent of s ∈ S ( k ) . Proof.
The cycle class of the specialization sp s (Λ X/S ) is the image of [Λ
X/S ] underthe Gysin map in lemma 2.3, and its degree is defined as the intersection numberof this image with the zero section X = X ֒ → W = T ∨ ( W/S ). As X and W areflat over S and X ֒ → W is a regular embedding, the degree is therefore constant by[16, th. 10.2] applied to the relative conormal variety V = Λ X/S . Note that V ֒ → W is not required to be a regular embedding; in order to apply loc. cit. we only needthat its base change to X is proper over S , which is true. (cid:3) Remark 2.5.
In characteristic p > the specialization of a family of conormalvarieties need not be a sum of conormal varieties, see [24, p. 215] . Similarly, weneed dim( S ) = 1 , otherwise we would have to restrict the class of morphisms asin [32] . For instance, for S = C ∋ s = (0 , we have i − s (Λ X/S ) = T ∨ ( W/S ) | X s for the subvariety X = { (( x, y, z ) , ( x − y z, y )) | ( x, y, z ) ∈ C } ⊂ W = C × S. Jump loci for the degree
Let f : W → S be a smooth dominant morphism from a smooth variety to asmooth curve as above. For any S -flat subvariety X ⊂ W and s ∈ S ( k ) we haveseen that sp s (Λ X/S ) − Λ X s ≥ , where the inequality means that the cycle on theleft hand side is effective or zero. It is natural to ask for which s ∈ S ( k ) the aboveinequality is strict. In the notation of lemma 2.3 this happens iff m Z > Z ⊆ Sing( X s ). The following provides a sufficient criterion for this to happenfor families of divisors: Proposition 3.1.
Assume d = codim( X, W ) = 1 . If
Sing(
X/S ) has an irreduciblecomponent which is contained in the fiber over some point s ∈ S ( k ) , then for thiscomponent Z ⊂ X s we have sp s (Λ X/S ) − Λ X s ≥ Λ Z . We divide the proof in several steps. Most of the argument works in arbitrarycodimension d , so for the moment we do not yet assume d = 1. It will be enough toprove the claim over some open dense subset of W . Fixing a general point p ∈ Z ( k )and working locally near that point, we can assume that • Z = Sing( X/S ) (equality as a scheme), • T ∨ ( W/S ) ≃ W × V is the trivial bundle with fiber V = T ∨ p ( W s ), • X ⊂ W is cut out by a regular sequence f , . . . , f d ∈ H ( W, O W ).By the first item each x ∈ ( X \ Z )( k ) is a smooth point of X t for t = f ( x ). Fixinga trivialization as in the second item, we can furthermore identify the conormalspace to X t ⊂ W t at x with a subspace in V = T ∨ x ( W t ) of codimension d . Considerthe relative Gauss map X \ Z −→ Gr( d, V )which sends each point to the corresponding conormal space. This is a rationalmap whose locus of indeterminacy is precisely Z . Let γ X : ˆ X → Gr( d, V ) denote its
EMICONTINUITY OF GAUSS MAPS 9 resolution of indeterminacy which is obtained by blowing up the base locus Z ⊂ X as in [16, sect. 4.4]: ˆ XX Gr( d, V ) π X γ X We want to control the image of the exceptional divisor E X = π − X ( Z ) ⊂ ˆ X underthe map α X = ( π X , γ X ) : ˆ X → X × Gr( d, V ) ⊂ W × Gr( d, V ) = Gr( d, T ∨ ( W/S )) . Lemma 3.2.
The morphism α X is a closed embedding.Proof. By assumption X ⊂ W is cut out by a regular sequence f , . . . , f d . Thesame then holds for each fiber X t ⊂ W t . Hence it follows that the relative singularlocus Sing( X/S ) is cut out as a closed subscheme of W by f , . . . , f d and by the d × d minors J ( i , . . . , i d ) := det ∂ i ( f ) · · · ∂ i ( f d )... . . . ... ∂ i d ( f ) · · · ∂ i d ( f d ) with 1 ≤ i < · · · < i d ≤ n , where we fix an arbitrary basis ∂ , . . . , ∂ n ∈ V ∨ forthe fiber of the relative tangent bundle and regard the basis vectors as relativederivations for the smooth morphism W → S .Now let ι : X × Gr( d, V ) ֒ → X × P (Λ d V ) be the Pl¨ucker embedding of theGrassmannian as a closed subvariety of projective space. We want to show that thecomposite β X := ι ◦ α X : ˆ X −→ X × Gr( d, V ) −→ X × P ( V d V )is a closed embedding. For this let I E O X be the ideal sheaf of Z ⊂ X . Then wehaveˆ X = Proj X R I for the graded Rees algebra R I := M n ≥ I n · t n ⊂ O X [ t ] , where t is a dummy variable to keep track of degrees. The homomorphism β ∗ X : O X ⊗ Sym • d ^ V ∨ ! −→ R I = M n ≥ I n · t n of graded O X -algebras satisfies β ∗ X (1 ⊗ ( ∂ i ∧ · · · ∧ ∂ i d )) = J ( i , . . . , i d ) | X · t ∈ I · t for 1 ≤ i < · · · < i k ≤ n . But we have seen above that the O X -module I isgenerated by the minors on the right hand side. Hence it follows that β ∗ X is anepimorphism in all degrees and so β X is a closed immersion. (cid:3) Remark 3.3. If d = 1 , then Gr( d, V ) = P V and the closed embedding α X inducesan isomorphism α X : ˆ X ∼ −→ P Λ X/S . Indeed, the blowup ˆ X is again reduced and irreducible by [21, II.7.16] . Via α X it istherefore an integral closed subscheme of W × P V and as such it can be recovered asthe Zariski closure of its restriction over the open dense subset S \ { s } ⊂ S , whereit coincides with P Λ X/S ⊂ W × P V by definition. In particular, one may look at the scheme-theoretic fiber of ˆ X over s ∈ S ( k ) tocompute the multiplicities in sp s (Λ X/S ). For d > d, , V ) −→ Gr( d, V )from the partial flag variety. This projection is a smooth equidimensional morphismof relative dimension d −
1. On the fiber product Y = ˆ X × Gr( d,V ) Fl( d, , V ) considerthe morphism α Y = ( π Y , γ Y ) : Y −→ X × P V where π Y : Y → ˆ X → X and γ Y : Y → Fl( d, , V ) → P V are the natural compositemaps. Taking the preimage of the previous exceptional divisor E X ⊂ ˆ X we get thefollowing lower bound on the specialization: Lemma 3.4.
The preimage E Y = π − Y ( Z ) ⊂ Y has dimension dim( E Y ) = n andsatisfies α Y ( E Y ) ⊂ P (Supp(sp s (Λ X/S ))) . Proof.
The statement about the dimension holds because dim( Y ) = n = dim( V )and since the subvariety E Y ⊂ Y is a divisor, being the preimage of the exceptionaldivisor E X ⊂ ˆ X under the fibration Y → ˆ X . To understand why the image γ Y ( E Y )is contained in the specialization, recall that f : X → S is smooth over the opensubset S ∗ = S \ { s } . The identifications Y ∗ := S ∗ × S Y ≃ S ∗ × S P Λ X/S ⊂ S ∗ × P T ∨ ( W/S )give the following Cartesian diagram where the vertical arrows are open embeddingsand the top horizontal arrow is a closed immersion: Y ∗ S ∗ × S P T ∨ ( W/S ) Y P T ∨ ( W/S )Now ˆ X is irreducible as a blowup of an irreducible variety. So Y is irreducible aswell, hence equal to the Zariski closure of its nonempty open subset Y ∗ ⊂ Y . Butthen P (sp s (Λ X/S )) = (cid:0) closure of S ∗ × S P Λ X/S in P T ∨ ( W/S ) (cid:1) s = (cid:0) closure of the image of Y ∗ in P T ∨ ( W/S ) (cid:1) s ⊇ α Y ( Y s )and we are done because by construction we have E Y ⊆ Y s . (cid:3) Corollary 3.5. If α Y : E Y → X × P V is generically finite onto its image, then wehave Λ Z ⊂ Supp(sp s (Λ X/S )) . EMICONTINUITY OF GAUSS MAPS 11
Proof.
Each irreducible component of P (Supp(sp s (Λ X/S ))) is the projectivizationof some conormal variety. Each of them has dimension n −
1, so lemma 3.4 andour generic finiteness assumption imply that α Y ( E Y ) must appear as one of thecomponents. But then this component is Λ Z because it maps onto Z ⊂ X . (cid:3) Note that by lemma 3.2 the morphism Y → X × Fl( d, , V ) is a closed immersionand hence generically finite onto its image. So corollary 3.5 finishes the proof ofproposition 3.1 since for codimension d = 1 the morphism Fl( d, , V ) → P V is anisomorphism. This is the only point where we use d = 1. For higher codimensionthe morphism α Y : E Y → X × P V is not always a closed embedding, as the followingexample shows, but it may still be generically finite onto its image as needed forcorollary 3.5: Example 3.6.
Let W = Spec k [ x, y, z, s ] → S = Spec k [ s ], and consider the familyof subvarieties X = { f = g = 0 } ⊂ W for ( f = x + y + s,g = x + z − s. Here Z = Sing( X/S ) ⊂ X is a fat point with ideal sheaf I = ( xy, xz, yz ) E O X andlooking at the minors of the Jacobian matrix we see that the relative Gauss mapis given in Pl¨ucker coordinates on the Grassmannian Gr(2 , V ) = Proj k [ w , w , w ]by γ X : X \ Z −→ Gr(2 , V ) = P , ( x, y, z, s ) [ w : w : w ] = [ yz : xz : − xy ] . Note that the right hand side does not involve the parameter s . Furthermore, wehave (2 x + y + z ) | X = ( f + g ) | X = 0 and hence the relative Gauss map γ X factorsover Q X = { w w + w w + w w = 0 } ⊂ Gr(2 , V ) . Write P V = Proj k [ v , v , v ] for the dual coordinates v i where the flag variety isgiven by Fl(2 , , V ) = { v w + v w + v w = 0 } ⊂ Gr(2 , V ) × P V then for Q Y = { w w + w w + w w = v w + v w + v w = 0 } ⊂ Gr(2 , V ) × P V. we get the following diagram where the squares are Cartesian and the hooked arrowsare closed immersions: E Y Y X × Q Y X × Fl(2 , , V ) X × P VE X ˆ X X × Q X X × Gr(2 , V )The composite of the arrows in the top row is the morphism α Y : E Y → X × P V that we are interested in. The diagram shows that it is not a closed immersion,over Z red = { } ⊂ X we have the factorization E red Y P VQ Yα Y where Q Y → P V is an irreducible cover of generic degree four! However, sincethe left diagonal arrow is a closed immersion and hence birational for dimensionreasons, the morphism α Y : E Y → X × P V is generically finite over its image.4. Generalities about families of rational maps
Before we apply the above to Gauss maps, let us recall some generalities aboutfamilies of rational maps. Let f : X → S be a faithfully flat morphism of varietiesof relative dimension n with irreducible fibers. Let L ∈ Pic ( X ) be a line bundleand V a rank n +1 vector subbundle of f ∗ L . Then for each point s ∈ S ( k ) we get alinear series V s ⊂ H ( X s , L s ) and we denote by φ s : X s P V s the correspondingrational map. Note that since the source and the target of this map have the samedimension, the map is a generically finite cover iff it is dominant. So we considerthe degree map deg : S ( k ) −→ N , s deg( φ s ) , where we put deg( φ s ) = 0 if φ s is not dominant. Lemma 4.1.
The degree map deg is constructible.Proof.
By [16, prop. 4.4] we can compute the degree in terms of Segre and Chernclasses as deg( φ s ) = Z X b c ( L s ) n − Z B s c ( L s ) n ∩ s ( B s , X s )where B s ⊂ X s denotes the base locus of the linear series V s ⊂ H ( X s , L s ). Wecan put together all these fiberwise base loci into a relative base locus and considerthe flattening stratification of this relative base locus. This is a stratification of S such that on each stratum the above intersection number is constant, hence thefunction deg is constructible. (cid:3) However, in general the degree is neither upper nor lower semi-continuous, asthe following variation of [11, ex. 2.3] shows:
Example 4.2.
Let S be a smooth affine curve with two marked points s ± ∈ S ( k )and fix positive integers n ± ≤ n <
27. Let p j : S −→ P for j = 1 , . . . , n be suchthat • for t = s ± the points p j ( t ) are in general position, • for t = s ± they consist of n ± general points on a given line ℓ and n − n ± points in general position not on that line.For t ∈ S ( k ) let f t : P P be the generically finite rational map defined by alinear system of four generic cubics passing trough the p j ( t ). By loc. cit. its degreeis deg( f t ) = ( − n for t = s ± , − ( n − n ± ) for t = t ± and n ± ≥ , since in the second case the indeterminacy locus of f ± consists of the chosen line ℓ together with the remaining n − n ± points. So the degree is a constructible functionas predicted by lemma 4.1. However, taking for example ( n, n + , n − ) = (20 , , EMICONTINUITY OF GAUSS MAPS 13 Gauss maps on abelian varieties
We now apply the above to an abelian scheme f : A → S , so in this section wetake W = A . Let X ⊂ A be a closed subvariety which is flat over S . For s ∈ S ( k )we have the Gauss map γ X s : P Λ X s −→ P V where V = H ( A s , Ω A s ). If this map is dominant, then for dimension reasons itis a generically finite cover and we denote by deg( γ X s ) its generic degree. If themap is not dominant we put deg( γ X s ) = 0; this happens iff the subvariety X s ⊂ A s is negligible in the sense that it arises by pull-back from some smaller dimensionalabelian quotient variety [37]. The degree of the above Gauss map is related to thedegree of conormal varieties as follows: Lemma 5.1.
We have deg(Λ X s ) = deg( γ X s ) .Proof. The degree of our Gauss map γ X s : P Λ X s → P V coincides with the degreeconsidered by Franecki and Kapranov terms of tangent rather than cotangent spacesin [15, sect. 2]: Up to the duality Gr( d, V ) ≃ Gr( g − d, V ∨ ) they study the map p ◦ q defined by the diagram e X s F ( d, , V ) G (1 , V ) = P V Sm( X s ) G ( d, V ) q p where e X s = Sm( X s ) × G ( d,V ) F ( d, g − , V ). By construction e X s ⊂ P Λ X s is an opensubset of the conormal variety, and their map p ◦ q is the restriction of our Gaussmap to this open subset. Hence the claim follows from [15, prop. 2.2]. (cid:3) In particular deg(Λ X s ) ≥
0. Together with the preservation of the total degreeunder Lagrangian specialization this leads to our first semicontinuity result:
Corollary 5.2.
The map S ( k ) → N , s deg( γ X s ) is lower semicontinuous.Proof. By Lemma 4.1, we know that the map is constructible. We have to showthat its values decreases under specialization. For this we may assume that S is acurve, and after base change to its normalization we may assume this curve to besmooth. Let d be the value from proposition 2.4. With notations as in lemma 2.3then deg(Λ X s ) = ( d if s / ∈ Σ ,d − δ ( s ) if s ∈ Σ , where δ ( s ) = P Z ⊂ X s m Z · deg(Λ Z ) with multiplicities m Z ≥
0. Now in the case ofabelian varieties the occuring degrees coincide with the degrees of the correspondingGauss maps by lemma 5.1. In particular, since the degrees of Gauss maps areobviously nonnegative, we have deg( γ Z ) ≥ δ ( s ) ≥
0, which provesthat the degree of the Gauss map is constant on an open dense subset and can onlydrop on the finitely many points of the complement. (cid:3)
To see where the function in the previous corollary actually jumps, recall thata subvariety Z ⊂ A s is negligible iff deg( γ Z ) = 0. Thus we obtain the followingsufficient jumping criterion: Corollary 5.3.
Suppose dim( S ) = 1 . Let X ⊂ A be a divisor which is flat over S and let ∈ S ( k ) be a point such that Sing( X ) has an irreducible component Z which is not negligible and not in the closure of S t =0 Sing( X t ) . Then there existsan open dense subset U ⊂ S such that deg( γ X t ) − deg( γ X ) ≥ deg( γ Z ) > for all t ∈ U ( k ) \ { } . Proof.
The first inequality follows from prop. 3.1, the second from our assumptionthat the subvariety Z ⊂ A is not negligible. (cid:3) Application to the Schottky problem
The moduli space A g of principally polarized abelian varieties of dimension g over the field k admits the finite filtration · · · ⊆ G d ⊆ G d − ⊂ · · · ⊆ G g ! = A g bythe Gauss loci G d := { ( A, Θ) ∈ A g | deg( γ Θ ) ≤ d } ⊆ A g . Our semicontinuity result implies:
Corollary 6.1.
For any d ∈ N the Gauss loci G d are closed in A g .Proof. The moduli space A g has a finite cover by a smooth quasi-projective varietyover which there exists universal theta divisor. On this cover the Gauss maps fittogether in a family of rational maps as in the setting of section 4. The Gaussloci are the level sets of the degree map, and we have to show that this map islower-semicontinuous. This follows from corollary 5.2. (cid:3) Using our sufficient criterion for jumps in the degree of Gauss maps, we canmoreover show that the stratification by the Gauss loci refines the stratification bythe Andreotti-Mayer loci N c = { ( A, Θ) ∈ A g | dim Sing(Θ) ≥ c } from [3]. We have to be a bit careful since negligible components are not seen bythe degree of the Gauss map, though we do not know whether such componentscan occur in the singular locus of theta divisors on indecomposable ppav’s: Question 6.2.
Does there exist an indecomposable ppav ( A, Θ) ∈ A g with a thetadivisor for which the singular locus Sing(Θ) ⊂ A has an irreducible componentwhose underlying reduced subscheme is negligible?In any case, the next corollary of our jumping criterion covers all those stratawhose general point is a simple abelian variety: Corollary 6.3.
Let c ∈ N , and let N ⊂ N c be an irreducible component whosegeneral point is a ppav whose singular locus of the theta divisor has no negligiblecomponents. Then N is an irreducible component of G d for some d ∈ N .Proof. Let s ∈ N ( k ) be a general point on the given component. Since the modulispace of ppav’s is a quasiprojective variety, we may pick an affine curve S ⊂ A g such that S ∩ N c = { s } and S meets N transversely. After passing to a finite coverwe may assume that there exists an abelian scheme f : A → S and a universaltheta divisor Θ ⊂ A over this curve, and by our choice of the curve we havedim Sing(Θ t ) < dim Sing(Θ s ) for all t = s. EMICONTINUITY OF GAUSS MAPS 15
Hence Sing(Θ s ) has an irreducible component not in the closure of S t = s Sing(Θ t )and so deg( γ Θ t ) > deg( γ Θ s ) for all t = s by corollary 5.3. Varying s in an open subset of the component N and varying S among all curves meeting this component transversely in the chosen point, we getthat some nonempty open subset of N is also an open subset of a Gauss locus G d for some d ∈ N . Hence the result follows by passing the the closure, since both theAndreotti-Mayer loci and the Gauss loci are closed in A g . (cid:3) As an application we get that the stratification by the degree of the Gauss mapgives a solution to the Schottky problem as conjectured in [11], where for the Prymversion we denote by D ( g ) the degree of the varieties of quadrics in P g − of rankat most three: Corollary 6.4.
We have the following components of Gauss loci in A g :(a) The locus of Jacobians is a component of G d for d = (cid:0) g − g − (cid:1) .(b) The locus of hyperelliptic Jacobians is a is a component of G d for d = 2 g − .(c) The locus of Prym varieties is a is a component of G d for d = D ( g ) + 2 g − .Proof. The locus of Jacobians is a component of an Andreotti-Mayer locus by [3],and by [30, Proposition 3.4] a general Jacobian variety is a simple abelian varietyand thus in particular has no negligible subvarieties other than itself. Furthermore,it is well-known that the degree of the Gauss map of a Jacobian is d = (cid:0) g − g − (cid:1) ,see e.g. [2, proof of prop. 10]. Hence part (a) follows from corollary 6.3. If wereplace Jacobians by hyperelliptic Jacobians, the above arguent works also in thehyperelliptic case, with the same references. For Prym varieties the argument isagain the same but now one has to replace reference [3] with [13], and reference [2]with [36, Main Theorem]. In the last reference the reader can also find an explicitexpression for the number D ( g ). (cid:3) A topological view on jump loci
In this section we work over the complex numbers with the Euclidean topology.Let W be a smooth complex projective variety. For a closed subvariety X ⊂ W the singular locus Sing( X ) and the conormal degree deg(Λ X ) are not topologicalinvariants of the subvariety, as example 1.5 shows. But both are related to theintersection cohomology IH • ( X ) which only depends on the homeomorphism typeof the subvariety in the Euclidean topology; see [6, 19, 20, 23, 29]. The Eulercharacteristic χ IC ( X ) = X i ≥ ( − i +dim( X ) dim IH i ( X )can be read off from a generalization of the Gauss-Bonnet theorem: The Kashiwaraindex formula [18, th. 9.1] writes it as a degree in the sense of definition 1.4. Moreprecisely χ IC ( X ) = deg(CC( δ X )) , where δ X ∈ Perv( W ) denotes the perverse intersection complex of X ⊆ W [5, 12]and where the characteristic cycle CC( δ X ) ∈ L ( W ) is an effective conic Lagrangian cycle which contains Λ X as a component of multiplicity one but may also have ascomponents the conormal varieties to certain Z ⊆ Sing( X ). Passing from conormalvarieties to characteristic cycles restores topological invariance of the degree: Example 7.1. In W = P the conormal degree for a smooth rational curve differsfrom the one for a cuspical cubic, see example 1.5. But this is compensated by adifference inCC( δ X ) = ( Λ X if X is a smooth rational curve , Λ X + Λ { p } if X is a cuspidal cubic with cusp p. which in both cases gives the total degree deg(CC( δ X )) = − X → S of complexvarieties the intersection cohomology Euler characteristic of the fibers varies. Basicstratification theory implies: Lemma 7.2.
The map s χ IC ( X s ) is constructible.Proof. The homeomorphism invariance of intersection cohomology [20, th. 4.1]shows that for a topologically locally trivial fibration over a connected base, allfibers have the same intersection cohomology. Any morphism of complex algebraicvarieties restricts to a topologically locally trivial fibration over a Zariski open densesubset of the target [35, cor. 5.1], so we are done by Noetherian induction. (cid:3)
In general the map in this lemma is neither upper nor lower semicontinuous, thejumps may go in both directions:
Example 7.3.
Let Q ⊂ P be a quadric. Then by [29, ex. 2.3.21 and th. 2.4.6] wehave χ IC ( Q ) = Q ≃ P × P is smooth , Q is a cone over a smooth rational curve , Q is a union of two projective planes . So for a family of quadrics whose general member is smooth, the number χ IC ( Q )jumps down on nodal quadrics but jumps up on reducible quadrics.Note that here the size of the jumps is precisely the Euler characteristic of thesingular locus. This fits with the following sheaf-theoretic version of the Lagrangianspecialization principle: Theorem 7.4.
Let f : W → S be a smooth proper family over a curve S , andlet X ⊂ W be a closed subvariety such that the morphism f : X → S is flat withgenerically reduced fibers. Then there exists d ∈ Z and a finite subset Σ ⊂ S suchthat χ IC ( X s ) = ( d for s ∈ S \ Σ ,d − deg(Λ( s )) for s ∈ Σ , where Λ( s ) = sp s (CC( δ X )) − CC( δ X s ) ∈ L ( W s ) is an effective cycle.Proof. We interpret Lagrangian specialization via perverse sheaves. For s ∈ S ( C )one has the functor of nearby cycles Ψ s : Perv( W ) → Perv( W s ), which is an exactfunctor with CC(Ψ s ( P )) = sp s (CC( P )) for all P ∈ Perv( W ) EMICONTINUITY OF GAUSS MAPS 17 by [18, th. 5.5]. Here we abuse notation and view CC( P ) as an element of thegroup of relative conic Lagrangian cycles L ( W/S ) via remark 2.1, discarding anycomponent that is not flat over S . The last part of the specialization lemma 2.3 hasa sheaf-theoretic version: For any closed S -flat subvariety X ⊂ W such that themap f : X → S has generically reduced fibers, there exists a finite subset Σ ⊂ S such that the semisimplification (Ψ s ( δ X )) ss of the perverse sheaf Ψ s ( δ X ) has theform (Ψ s ( δ X )) ss ≃ ( δ X s for s ∈ S \ Σ ,δ X s ⊕ P ( s ) for s ∈ Σ , where P ( s ) ∈ Perv( X s ) is a perverse sheaf with support contained in Sing( X s ). Sowe get sp s (CC( δ X )) = CC(Ψ s ( δ X )) = ( CC( δ X s ) for s ∈ S \ Σ , CC( δ X s ) + Λ( s ) for s ∈ Σ , where Λ( s ) = CC( P ( s )) ∈ L ( W s ) is effective, being the characteristic cycle of aperverse sheaf. Hence the result follows by noting that if f : W → S is proper,then by proposition 2.4 the degree d = deg(sp s (CC( δ X ))) is independent of s . (cid:3) In the case of abelian varieties the positivity of conormal degrees then gives ananalog of theorem 1.7. The same argument works for a much wider class of varieties,we only need the following positivity property:
Definition 7.5.
A variety X satisfies the signed Euler characteristic property if wehave χ ( X, P ) := X i ∈ Z ( − i dim H i ( X, P ) ≥ P ∈ Perv( X ) . The terminology is borrowed from [14]. The above property holds for semiabelianvarieties [15] and hence also for any finite cover of closed subvarieties of them:
Lemma 7.6.
If a variety A has the signed Euler characteristic property, then sodoes any variety with a finite morphism to A . In particular, any variety with a finitemorphism to a semiabelian variety has the signed Euler characteristic property.Proof. If f : X → A is a finite morphism, then for any perverse sheaf P ∈ Perv( X )the direct image is a perverse sheaf Rf ∗ ( P ) ∈ Perv( A ). If A has the signed Eulercharacteristic property, which holds for instance for abelian varieties [15], then weget χ ( A, P ) = χ ( A, Rf ∗ ( P )) ≥ (cid:3) The above theorem shows that for any family of such varieties the intersectioncohomology Euler characteristic is semicontinuous:
Corollary 7.7.
Let f : W → S be a smooth proper morphism to a variety S , andlet X ⊂ W be a closed subvariety such that f : X → S is flat and all its fibersare generically reduced and have the signed Euler characteristic property. Then foreach d ∈ N the subsets S d = (cid:8) s ∈ S | χ IC ( X s ) ≤ d (cid:9) ⊆ S are Zariski closed.Proof. We must show that χ IC ( X s ) cannot increase under specialization. For thiswe can assume S is a smooth curve. Then theorem 7.4 applies, here χ ( X s , P ( s )) ≥ X s has the signed Euler characteristic property. (cid:3) In deciding where the Euler characteristic actually jumps, we need to be morecareful. Proposition 3.1 gives a way to see extra components in sp s (CC( δ X )) butdoes not guarantee that these enter in a new summand Λ( s ), a priori they couldalso appear in CC( δ X s ); however, this second case can only happen if CC( δ X s ) isreducible, which one can often exclude by a direct computation.Let us illustrate this again with theta divisors. Corollary 7.7 says that for d ∈ N the loci X d = { ( A, Θ) ∈ A g | χ IC (Θ) ≤ d } ⊆ A g . are closed, and by the homeomorphism invariance of intersection cohomology theyonly depend on the topology of the theta divisor. This provides a topological viewon Andreotti-Mayer loci, for instance: Corollary 7.8.
Let N ⊂ N c be an irreducible component of an Andreotti-Mayerlocus such that a general point of this component is a ppav ( A, Θ) with the propertythat • CC( δ Θ ) is irreducible, and • Sing(Θ) has no negligible components.Then N is also an irreducible component of X d for some d ∈ N .Proof. Use the same argument as in corollary 6.3, together with the remark afterthe proof of corollary 7.7. (cid:3)
This in particular applies to the locus of Jacobians. In the following corollary wedo not mention hyperelliptic Jacobians because for them CC( δ Θ ) is reducible, andwe haven’t checked what happens for a generic Prym variety. However, we includethe Andreotti-Mayer locus N ⊂ A g of ppav’s with a singular theta divisor: Corollary 7.9.
Inside the moduli space A g we have:(1) The locus N is equal to X d for d = ( g ! − if g is odd ,g ! − if g is even . (2) The locus of Jacobians is a component of X d for d = (cid:0) g − g − (cid:1) .Proof. (1) By definition A g \ N consists of all ppav’s ( A, Θ) with a smooth thetadivisor and for those we know that χ IC (Θ) = g ! because for a smooth varietyintersection cohomology equals Betti cohomology. But at a generic point ( A, Θ) ofeach of the two components of N the theta divisor has one respectively two nodes,and then χ IC ( δ Θ ) = ( g ! − k if g is even ,g ! − k if g is odd , where k ∈ { , } denotes the number of nodes [26, proof of prop. 4.2(2)]. Hence theclaim follows by corollary 7.7. Note that the degree of the classical Gauss map isdeg(Λ Θ ) = g ! − k in both cases, but for odd g the cycle CC( δ Θ ) = Λ Θ + Λ Sing(Θ) is reducible and we cannot directly apply corollary 7.8.(2) For Jacobians of nonhyperelliptic curves we know that CC( δ Θ ) = Λ Θ isirreducible by [7, th. 3.3.1], so if we specialize to such a Jacobian, then any newcomponent of the specialization must enter in Λ( s ). So the same argument as inthe proof of corollary 6.4 shows that the locus of Jacobians is a component of X d EMICONTINUITY OF GAUSS MAPS 19 where d = χ IC (Θ) = deg(CC( δ Θ )) = deg(Λ Θ ) is the degree of the Gauss map forthe theta divisor on a general Jacobian variety as in corollary 6.4. (cid:3) The above is still only a weak solution to the Schottky problem, though χ IC (Θ)also appears as the dimension of an irreducible representation of a certain reductivegroup which gives more information [25, sect. 4]. The following example for g = 4illustrates the difference between the various numerical invariants: Example 7.10.
Let ( A, Θ) ∈ A . • If Sing(Θ) consists of 8 nodes, then deg( γ Θ ) = χ IC ( δ Θ ) = 8. • If Sing(Θ) consists of 5 nodes, then deg( γ Θ ) = χ IC ( δ Θ ) = 14. • If ( A, Θ) is a hyperelliptic Jacobian, then deg( γ Θ ) = 8 and χ IC ( δ Θ ) = 14.So there are non-homeomorphic theta divisors whose Gauss maps have the samedegree. Are there also homeomorphic theta divisors with different Gauss degrees? References
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