Andreotti–Mayer Loci and the Schottky Problem

We prove a lower bound for the codimension of the Andreotti-Mayer locus N_{g,1} and show that the lower bound is reached only for the hyperelliptic locus in genus 4 and the Jacobian locus in genus 5. In relation with the boundary of the Andreotti-Mayer loci we study subvarieties of principally polarized abelian varieties (B,Theta) parametrizing points b such that Theta and the translate Theta_b are tangentially degenerate along a variety of a given dimension.


Introduction
The Schottky problem asks for a characterization of Jacobian varieties among all principally polarized abelian varieties. In other words, it asks for a description of the Jacobian locus J g in the moduli space A g of all principally polarized abelian varieties of given dimension g. In the 1960's Andreotti and Mayer (see [2]) pioneered an approach based on the fact that the Jacobian variety of a non-hyperelliptic (resp. hyperelliptic) curve of genus g ≥ 3 has a singular locus of dimension g − 4 (resp. g − 3). They introduced the loci N g,k of principally polarized abelian varieties (X, Θ X ) of dimension g with a singular locus of Θ X of dimension ≥ k and showed that J g (resp. the hyperelliptic locus H g ) is an irreducible component of N g,g−4 (resp. N g,g−3 ). However, in general there are more irreducible components of N g,g−4 so that the dimension of the singular locus of Θ X does not suffice to characterize Jacobians or hyperelliptic Jacobians. The locus N g,0 of abelian varieties with a singular theta divisor has codimension 1 in A g and in a beautiful paper (see [27]) Mumford calculated its class. But in general not much is known about these Andreotti-Mayer loci N g,k . In particular, we do not even know their codimension. In this paper we give estimates for the codimension of these loci. These estimates are in general not sharp, but we think that the following conjecture gives the sharp bound.
Conjecture 1.1. If 1 ≤ k ≤ g − 3 and if N is an irreducible component of N g,k whose general point corresponds to an abelian variety with endomorphism ring Z then codim Ag (N ) ≥ k+2 2 . Moreover, equality holds if and only if one of the following happens: (i) g = k + 3 and N = H g ; (ii) g = k + 4 and N = J g .
We give some evidence for this conjecture by proving the case k = 1. In our approach we need to study the behaviour of the Andreotti-Mayer loci at the boundary of the compactified moduli space. A principally polarized (g − 1)dimensional abelian variety (B, Ξ) parametrizes semi-abelian varieties that are extensions of B by the multiplicative group G m . This means that B maps to a part of the boundary of the compactified moduli spaceÃ g and we can intersect B with the Andreotti-Mayer loci. This motivates the definition of loci N k (B, Ξ) ⊂ B for a principally polarized (g − 1)-dimensional abelian variety (B, Ξ). They are formed by the points b in B such that Ξ and its translate Ξ b are 'tangentially degenerate' (see Section 11 below) along a subvariety of dimension k. These intrinsically defined subvarieties of an abelian variety are interesting in their own right and deserve further study. The conjecture above then leads to a boundary version that gives a new conjectural answer to the Schottky problem for simple abelian varieties. (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.
In our approach we will use a special compactificationÃ g of A g (see [29,28,5]). The points of the boundary ∂Ã g =Ã g − A g correspond to suitable compactifications 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.

The universal theta divisor
Let π : X g → A g be the universal principally polarized abelian variety of relative dimension g over the moduli space A g of principally polarized abelian varieties of dimension g over C. In this paper we will work with orbifolds and we shall identify X g (resp. A g ) with the orbifold Sp(2g, Z) ⋉ Z 2g \H g × C g (resp. with Sp(2g, Z)\H g ), where H g = {(τ ij ) ∈ Mat(g × g, C) : τ = τ t , Im(τ ) > 0} Documenta Mathematica 13 (2008)   is the usual Siegel upper-half space of degree g. The τ ij with 1 ≤ i ≤ j ≤ g are coordinates on H g and we let z 1 , ..., z g be coordinates on C g . The Riemann theta function ϑ(τ, z), given on H g × C g by is a holomorphic function and its zero locus is an effective divisorΘ on H g × C g which descends to a divisor Θ on X g . If the abelian variety X is a fibre of π, then we let Θ X be the restriction of Θ to X. Note that since θ(τ, z) satisfies θ(τ, −z) = θ(τ, z), the divisor Θ X is symmetric, i.e., ι * (Θ X ) = Θ X , where ι = −1 X : X → X is multiplication by −1 on X. The divisor Θ X defines the line bundle O X (Θ X ), which yields the principal polarization on X. The isomorphism class of the pair (X, Θ X ) represents a point ζ of A g and we will write ζ = (X, Θ X ). Similarly, it will be convenient to identify a point ξ of Θ with the isomorphism class of a representative triple (X, Θ X , x), where ζ = (X, Θ X ) represents π(ξ) ∈ A g and x ∈ Θ X . The tangent space to X g at a point ξ, with π(ξ) = ζ, will be identified with the tangent space T X,x ⊕T Ag,ζ ∼ = T X,0 ⊕Sym 2 (T X,0 ). If ξ = (X, Θ X , x) corresponds to the Sp(2g, Z) ⋉ Z 2g -orbit of a point (τ 0 , z 0 ) ∈ H g × C g , then the tangent space T Xg,ξ to X g at ξ can be identified with the tangent space to H g × C g at (τ 0 , z 0 ), which in turn is naturally isomorphic to C g(g+1)/2+g , with coordinates (a ij , b ℓ ) for 1 ≤ i, j ≤ g and 1 ≤ ℓ ≤ g that satisfy a ij = a ji . We thus view the a ij 's as coordinates on the tangent space to H g at τ 0 and the b ℓ 's as coordinates on the tangent space to X or its universal cover. An important remark is that by identifying the tangent space to A g at ζ = (X, Θ X ) with Sym 2 (T X,0 ), we can view the projectivized tangent space P(T Ag ,ζ ) ∼ = P(Sym 2 (T X,0 )) as the linear system of all quadrics in the dual of P g−1 = P(T X,0 ). In particular, the matrix (a ij ) can be interpreted as the matrix defining a dual quadric in the space P g−1 with homogeneous coordinates (b 1 : . . . : b g ). Quite naturally, we will often use (z 1 : . . . : z g ) for the homogeneous coordinates in P g−1 .
Recall that the Riemann theta function ϑ satisfies the heat equations ∂ ∂z i ∂ ∂z j ϑ = 2π √ −1(1 + δ ij ) ∂ ∂τ ij ϑ for 1 ≤ i, j ≤ g, where δ ij is the Kronecker delta. We shall abbreviate this equation as ∂ i ∂ j ϑ = 2π √ −1(1 + δ ij )∂ τij ϑ, where ∂ j means the partial derivative ∂/∂z j and ∂ τij the partial derivative ∂/∂τ ij . One easily checks that also all derivatives of θ verify the heat equations. We refer to [39] for an algebraic interpretation of the heat equations in terms of deformation theory. If ξ = (X, Θ X , x) ∈ Θ corresponds to the Sp(2g, Z) ⋉ Z 2g -orbit of a point (τ 0 , z 0 ), then the Zariski tangent space T Θ,ξ to Θ at ξ is the subspace of T Xg,ξ ≃ C g(g+1)/2+g defined, with the above conventions, by the linear equation in the variables (a ij , b ℓ ), 1 ≤ i, j ≤ g, 1 ≤ ℓ ≤ g. As an immediate consequence we get the result (see [36], Lemma (1.2)): Lemma 2.1. The point ξ = (X, Θ X , x) is a singular point of Θ if and only if x is a point of multiplicity at least 3 for Θ X .

The locus S g
We begin by defining a suborbifold of Θ supported on the set of points where π |Θ fails to be of maximal rank.
Lemma 2.1 implies that the support of S g is the union of Sing(Θ) and of the set of smooth points of Θ where π |Θ fails to be of maximal rank. Set-theoretically one has S g = {(X, Θ X , x) ∈ Θ : x ∈ Sing(Θ X )} and codim Xg (S g ) ≤ g + 1. It turns out that every irreducible component of S g has codimension g + 1 in X g (see [8] and an unpublished preprint by Debarre [9]). We will come back to this later in §7 and §8.
In this setting, the equation (1) for the tangent space to T Θ,ξ can be written as: (5) a · q t + b · ∂ϑ(τ 0 , z 0 ) t = 0 where ∂ denotes the gradient. Suppose now the point ξ = (X, Θ X , x) in S g , corresponding to (τ 0 , z 0 ) ∈ H g ×C g is not a point of Sing(Θ). By Lemma 2.1 the matrix M is not zero and therefore we can associate to ξ a quadric Q ξ in the projective space P(T X,x ) ≃ P(T X,0 ) ≃ P g−1 , namely the one defined by the equation b · M · b t = 0.
Recall that b = (b 1 , . . . , b g ) is a coordinate vector on T X,0 and therefore (b 1 : . . . : b g ) are homogeneous coordinates on P(T X,0 ). We will say that Q ξ is indeterminate if ξ ∈ Sing(Θ). The vector q naturally lives in Sym 2 (T X,0 ) ∨ and therefore, if q is not zero, the point [q] ∈ P(Sym 2 (T X,0 ) ∨ ) determines a quadric in P g−1 = P(T X,0 ). The heat equations imply that this quadric coincides with Q ξ . Consider the matrix defining the Zariski tangent space to S g at a point ξ = (X, Θ X , x). We denote by r := r ξ the corank of the quadric Q ξ , with the convention that r ξ = g if ξ ∈ Sing(Θ), i.e., if Q ξ is indeterminate. If we choose coordinates on C g such that the first r basis vectors generate the kernel of q then the shape of the matrix A of the system (3) is where q and q k are as above and B is a (g − r) × g-matrix with the first r columns 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 , x) of S g . Before stating the result, we need one more piece of notation. Given a non-zero vector If z = e i is the i-th vector of the standard basis, one writes ∂ i Q ξ = Q ξ,i instead of Q ξ,ei for i = 1, . . . , g. We will use similar notation for higher order derivatives or even for differential operators applied to a quadric.
Definition 3.2. We let Q ξ be the linear system of quadrics in P(T X,0 ) spanned by Q ξ and by all quadrics Q ξ,b with b ∈ ker(Q ξ ).
Since Q ξ has corank r, the system Q ξ is spanned by r+1 elements and therefore dim(Q ξ ) ≤ r. This system may happen to be empty, but then Q ξ is indeterminate, i.e., ξ lies in Sing(Θ). Sometimes we will use the lower suffix x instead of ξ to denote quadrics and linear systems, e.g. we will sometimes write Q x instead of Q ξ , etc. By the heat equations, the linear system Q ξ is the image of the vector subspace of Sym 2 (T X,0 ) ∨ spanned by the vectors q, q 1 , . . . , q r . Proposition 3.3. The subscheme S g is smooth of codimension g + 1 in X g at the point ξ = (X, Θ X , x) of S g if and only if the following conditions hold: ∈ Sing(Θ), i.e., Q ξ is not indeterminate and of corank r < g; (ii) the linear system Q ξ has maximal dimension r; in particular, if b 1 , . . . , b r span the kernel of Q ξ , then the r + 1 quadrics Q ξ , Q ξ,b1 , . . . , Q ξ,br are linearly independent.
Proof. The subscheme S g is smooth of codimension g + 1 in X g at ξ if and only if the matrix A appearing in (6) has maximal rank g + 1. Since the submatrix B of A has rank g − r, the assertion follows.
Corollary 3.4. If Q ξ is a smooth quadric, then S g is smooth at ξ = (X, Θ X , x).

Quadrics and Cornormal Spaces
Next we study the differential of the restriction to S g of the map π : X g → A g at a point ξ = (X, Θ X , x) ∈ S g . We are interested in the kernel and the image of dπ |Sg ,ξ . We can view these spaces in terms of the geometry of P g−1 = P(T X,0 ) as follows: is a linear subspace of P(T X,0 ) and 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 , x) be a point of S g . Then: (i) Π ξ is the vertex of the quadric Q ξ . In particular, if ξ is a singular point of Θ, then Π ξ is the whole space P(T X,0 ); (ii) Σ ξ contains the linear system Q ξ .
Proof. The assertions follow from the shape of the matrix A in (6).
This proposition tells us that, given a point ξ = (X, Θ X , x) ∈ S g , the map dπ |Sg ,ξ is not injective if and only if the quadric Q ξ is singular. The orbifold S g is stratified by the corank of the matrix (∂ i ∂ j θ).
Definition 4.2. For 0 ≤ k ≤ g we define S g,k as the closed suborbifold of S g defined by the equations on H g × C g (7) ϑ(τ, z) = 0, ∂ j ϑ(τ, z) = 0, (j = 1, . . . , g), Geometrically this means that ξ ∈ S g,k if and only if dim(Π ξ ) ≥ k − 1 or equivalently Q ξ has corank at least k. We have the inclusions and S g,1 is the locus where the map dπ |Sg ,ξ is not injective. The loci S g,k have been 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 g,k+1 . Then we have Proof. Locally, in a neighborhood U in S g of a point z of Z\S g,k+1 we have a morphism f : U → Q, where Q is the linear system of all quadrics in P g−1 .
The map f sends ξ = (X, Θ X , x) ∈ U to Q ξ . The scheme S g,k is the pullback of the subscheme Q k of Q formed by all quadrics of corank k. Since codim Q (Q k ) = k+1 2 , the assertion follows. Using the equations (7) it is possible to make a local analysis of the schemes S g,k , e.g. it is possible to write down equations for their Zariski tangent spaces (see §6 for the case k = g). This is however not particularly illuminating, and we will not dwell on this here. It is useful to give an interpretation of the points ξ = (X, Θ X , x) ∈ S g,k in terms of singularities of the theta divisor Θ X . Suppose that ξ is such that Sing(Θ X ) contains a subscheme isomorphic to Spec(C[ǫ]/(ǫ 2 )) supported at x. This subscheme of X is given by a homomorphism where ∆ (1) is a non-zero differential operator of order ≤ 1, hence ∆ (1) = ∂ b , for some non-zero vector b ∈ C g . Then the condition Spec(C[ǫ]/(ǫ 2 )) ⊂ Sing(Θ X ) is equivalent to saying that ϑ and ∂ b ϑ satisfy the equations and this, in turn, is equivalent to the fact that the quadric Q ξ is singular at More generally, we have the following proposition, which explains the nature of the points in S g,k for k < g.
Proposition 4.4. Suppose that x ∈ Sing(Θ X ) does not lie on Sing(Θ). Then Sing(Θ X ) contains a scheme isomorphic to Spec(C[ǫ 1 , . . . , ǫ k ]/(ǫ i ǫ j : 1 ≤ i, j ≤ k < g)) supported at x if and only if the quadric Q ξ has corank r ≥ k. Moreover, the Zariski tangent space to Sing(Θ X ) at x is the kernel space of Q ξ .
Proof. With a suitable choice of coordinates in X, the condition that the scheme Spec(C[ǫ 1 , . . . , ǫ k ]/(ǫ i ǫ j : 1 ≤ i, j ≤ k < g)) is contained in Sing(Θ X ) is equivalent to the fact that the functions ϑ and ∂ i ϑ for i = 1, . . . , k satisfy (8). But this the same as saying that ∂ i ∂ j ϑ(τ 0 , z 0 ) is zero for i = 1, . . . , k, j = 1, . . . , g, and the vectors e i , i = 1, . . . , k, belong to the kernel of Q ξ . This settles the first assertion. The scheme Sing(Θ X ) is defined by the equations (2), where τ is now fixed 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 X ) at x are . . , g i.e., b · M = 0, which proves the second assertion.

Curvi-linear subschemes in the singular locus of theta
supported at x is given by a homomorphism (9) δ with ∆ (j) a differential operator of order ≤ j, j = 1, . . . , N , with ∆ (N ) nonzero, and ∆ (0) (f ) = f (x). The condition that the map δ is a homomorphism is equivalent to saying that for any pair (f, g) of elements of O X,x . Two such homomorphisms δ and δ ′ define the same subscheme if and only if they differ by composition with a automorphism of C[t]/(t N +1 ).
Lemma 5.1. The map δ defined in (9) is a homomorphism if and only if there exist translation invariant vector fields D 1 , . . . , D N on X such that for every k = 1, . . . , N one has Moreover, two N -tuples of vector fields (D 1 , . . . , D N ) and (D ′ 1 , . . . , D ′ N ) determine 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 1 , . . . , c N , with c 1 = 0, such that Proof. If the differential operators ∆ (k) , k = 1, . . . , N , are as in (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.
is the homogeneous part of degree i, and write D k instead of D (k) 1 . Using (10) one verifies that for every k = 1, . . . , N and every positive i ≤ k one has Formula (11) follows by induction and easy combinatorics. To prove the final assertion, use the fact that an automorphism of In formula (11) . . , h i ℓ are positive integers. Thus formula (11) can be written as (12) ∆ where the subscript {h i1 , . . . , h i ℓ } means that the sum is taken over all ℓ-tuples Remark 5.2. Let x ∈ X correspond to the pair (τ 0 , z 0 ). The differential operators ∆ (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 0 , z 0 ) = 0.
We want now to express the conditions in order that a 0-dimensional curvilinear subscheme of X of length N + 1 supported at a given point x ∈ X corresponding to the pair (τ 0 , z 0 ) and determined by a given N -tuple of vector fields (D 1 , . . . , D N ) lies in Sing(Θ X ). To do so, we keep the notation we introduced above. Let us write D i = g ℓ=1 η iℓ ∂ ℓ , so that D i corresponds to the vector η i = (η i1 , . . . , η ig ). As before we denote by M the matrix (∂ i ∂ j θ(τ 0 , z 0 )).
Proposition 5.3. The 0-dimensional curvi-linear subscheme R of X of length N + 1, supported at the point x ∈ X corresponding to the pair (τ 0 , z 0 ) and determined by the N -tuple of vector fields (D 1 , . . . , D N ) lies in Sing(Θ X ) if and only if x ∈ Sing(Θ X ) and moreover for each k = 1, . . . , N one has where the sum is taken over all ℓ-tuples of positive integers (h i1 , . . . , h i ℓ ) with 1 ≤ i 1 < · · · < i ℓ ≤ k and i 1 h i1 + · · · + i ℓ h i ℓ = k.
Proof. The scheme R is contained in Sing(Θ X ) if and only if one has By Remark 5.2 this is equivalent to The assertion follows by the expression (12) of the operators ∆ (k) .
For instance, consider the scheme R 1 , supported at x ∈ Sing(Θ X ), corresponding to the vector field D 1 . Then R 1 is contained in Sing(Θ X ) if and only if This agrees with Proposition 4.4. If R 2 is the scheme supported at x and corresponding to the pair of vector fields (D 1 , D 2 ), then R 2 is contained in Sing(Θ X ) if and only if, besides (14) one has also Next, consider the scheme R 3 supported at x and corresponding to the triple of vector fields (D 1 , D 2 , D 3 ). Then R 3 is contained in Sing(Θ X ) if and only if, besides (14) and (15) one has also (16) (1/3!)η 1 · ∂ 2 η1 M + (1/2)η 2 · ∂ η1 M + η 3 · M = 0 and so on. Observe that (13) can be written in more than one way. For example η 2 · ∂ η1 M = η 1 · ∂ η2 M so that (16) could also be written as (1/3!)η 1 · ∂ 2 η1 M + (1/2)η 1 · ∂ η2 M + η 3 · M = 0. So far we have been working in a fixed abelian variety X. One can remove this restriction by working on S g and by letting the vector fields D 1 , . . . , D N vary with X, which means that we let the vectors η i depend on the variables τ ij . Then the equations (13) define a subscheme S g (D) of Sing(Θ) which, as a set, is the locus of all points ξ = (X, Θ X , x) ∈ S g such that Sing(Θ X ) contains a curvi-linear scheme of length N + 1 supported at x, corresponding to the N -tuple of vector fields D = (D 1 , . . . , D N ), computed on X. One can compute the Zariski tangent space to S g (D) at a point ξ = (X, Θ X , 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 1 = . . . = D N = 0, and call b the corresponding tangent vector to X at the origin, depending on the the variables τ ij . In this case we use the notation D b,N = (D 1 , . . . , D N ) and we denote by R x,b,N the corresponding curvi-linear scheme supported at x. For a given such D = (D 1 , . . . , D N ), consider the linear system of quadrics Σ ξ (D) = P(Im(dπ |Sg (D),ξ ) ⊥ ) in P(T X,0 ). One has again an interpretation of these quadrics in terms of the normal space: Proposition 5.4. In the above setting, the space Σ ξ (D b,N ) contains the quadrics Q ξ , ∂ b Q ξ , . . . , ∂ N b Q ξ . Proof. The equations (13) take now the form By differentiating the assertion immediately follows.

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 1 , . . . , i g ) with i 1 , . . . , i g non-negative integers we set z I = z i1 1 · · · z ig g and denote by ∂ I the operator ∂ i1 1 · · · ∂ ig g . Moreover, we let |I| = g ℓ=1 i ℓ , which is the length of I and equals the order of the operator ∂ I . Definition 6.1. For a positive integer r we let S (r) g be the subscheme of X g which is defined on H g × C g by the equations (17) ∂ I ϑ(τ, z) = 0, |I| = 0, . . . , r − 1.
One has the chain of subschemes x has multiplicity ≥ r for Θ X }. One denotes by Sing (r) (Θ X ) the subscheme of Sing(Θ X ) formed by all points of multiplicity at least r. One knows that S (r) g = ∅ as soon as r > g (see [37]). We can compute the Zariski tangent space to S (r) g at a point ξ = (X, Θ X , x) in the same vein, and with the same notation, as in §3. Taking into account that θ and all its derivatives verify the heat equations, we find the equations by replacing in (3) the term θ(τ 0 , z 0 ) by ∂ I θ(τ 0 , z 0 ). As in §3, we wish to give some geometrical interpretation. For instance, we have the following lemma which partially extends Lemma 2.1 or 3.3. Next we are interested in the differential of the restriction of the map π : X g → A g to S (r) g at a point ξ = (X, Θ X , x) which does not belong to S (r+1) g . This means that Θ X has a point of multiplicity exactly r at x. If we assume, as we may, that x is the origin of X , i.e. z 0 = 0, then the Taylor expansion of θ has the form where ϑ i is a homogeneous polynomial of degree i in the variables z 1 , . . . , z g and θ r = I=(i1,...,ig ),|I|=r is not identically zero. The equation θ r = 0 defines a hypersurface T C ξ of degree r in P g−1 = P(T X,0 ), which is the tangent cone to Θ X at x. We will denote by Vert(T C ξ ) the vertex of T C ξ , i.e., the subspace of P g−1 which is the locus of points of multiplicity r of T C ξ . Note that it may be empty. In case r = 2, the tangent cone T C ξ is the quadric Q ξ introduced in §3 and Vert(T C ξ ) is its vertex Π ξ . More generally, for every s ≥ r, one can define the subscheme T C of P g−1 = P(T X,0 ) defined by the equations which is called the asymptotic cone of order s to Θ X at x. Fix a multi-index J = (j 1 , . . . , j g ) of length r − 2. For any pair (h, k) with 1 ≤ h, k ≤ g, let J (h,k) be the multi-index of length r obtained from J by first increasing by 1 the index j h and then by 1 the index j k (that is, by 2 if they coincide). Consider then the quadric Q J ξ in P g−1 = P(T X,0 ) defined by the equation with the usual convention that the quadric is indeterminate if the left-hand-side is identically zero. This is a polar quadric of T C ξ , namely it is obtained from T C ξ by iterated operations of polarization. Moreover all polar quadrics are in the span Q J ξ , |J| = r − 2 . We will denote by Q for every non-zero vector b ∈ C g such that [b] ∈ Vert(T C ξ ).
We are now interested in the kernel and the image of dπ |S (r) g ,ξ . Equivalently we may consider the linear system of quadrics Σ (r) ξ = P(Im(dπ |S (r) g ,ξ ) ⊥ ), and the subspace Π (r) ξ = P(ker(dπ |S (r) g ,ξ )) of P(T X,0 ). The following proposition partly extends Proposition 4.1 and 4.4 and its proof is similar.
Remark 6.4. As a consequence, just like in Proposition 4.4, one sees that for ξ = (X, Θ X , x) the Zariski tangent space to Sing (r) (Θ X ) at x is contained in Vert(T C ξ ).
As an application, we have: Documenta Mathematica 13 (2008) 453-504 Proposition 6.5. Let ξ = (X, Θ X , x) be a point of a component Z of S (3) g such that T C ξ is not a cone. Then dim(Q (3) ξ ) = g − 1 and therefore the codimension of the image of Z in A g is at least g.
Proof. Since T C ξ is a not a cone its polar quadrics are linearly independent.
The following example shows that the above bound is sharp for g = 5.
Example 6.6. Consider the locus C of intermediate Jacobians of cubic threefolds in A 5 . Note that dim(A 5 ) = 15 and dim(C) = 10. These have (at least) an isolated triple point on their theta divisor whose tangent cone gives back the cubic threefold. The locus C is dominated by an irreducible component of S

The Andreotti-Mayer loci
Andreotti and Mayer consider in A g the algebraic sets of principally polarized abelian varieties X with a locus of singular points on Θ X of dimension at least k. More generally, we are interested in the locus of principally polarized abelian varieties possessing a k-dimensional locus of singular points of multiplicity r on the theta divisor. To define these loci scheme-theoretically we consider the morphism π : X g → A g and the quasi-coherent sheaf on A g F (r) Definition 7.1. For integers k and r with 0 ≤ k ≤ g − 2 and 2 ≤ r ≤ g we define N g,k,r as the support of F (r) k . We also set M g,k,r = π −1 (N g,k,r ), a subscheme of both S g,k and S (r) g . We write N g,k and M g,k for N g,k,2 and M g,k,2 .
The schemes N g,k are the so-called Andreotti-Mayer loci in A g , which were introduced in a somewhat different way in [2]. Note that N g,k,r is locally defined by an annihilator ideal and so carries the structure of subscheme. Corollary 8.12 below and results by Debarre [11] (see §19) imply that the scheme structure at a general point of N g,0 defined above coincides with the one considered by Mumford in [27]. We now want to see that as a set N g,k,r is the locus of points corresponding to (X, Θ X ) such that Sing(Θ X ) has an irreducible component of dimension ≥ k of points of multiplicity ≥ r for Θ X .
Lemma 7.2. Let X be an abelian variety of dimension g and W ⊂ X an irreducible reduced subvariety of dimension n and let ω W be its dualizing sheaf.
To see this, note that by [24], p. 48 ff (see also [17], Exerc. 6.10, p. 239, 7.2, p. 249), there exists a map which is injective as one sees by looking at the smooth part of W .
LetW be a desingularization of W ′ . According to [22] we would be the zero map contradicting the fact that W has dimension n.
Proof. By the previous lemma and Serre duality for a reduced irreducible sub- This implies the corollary.
There are the inclusions If p = (n 1 , . . . , n r ) with 1 ≤ n 1 ≤ . . . ≤ n r < g and n 1 + . . . + n r = g is a partition of g we write A g,p for the suborbifold (or substack) of A g corresponding to principally polarized abelian varieties that are a product of r principally polarized abelian varieties of dimensions n 1 , . . . , n r . We write r(p) = r for the length of the partition and write A g,[r] for the suborbifold ∪ r(p)=r A g,p of A g corresponding to pairs (X, Θ X ) isomorphic as a polarized abelian variety to the product of r principally polarized abelian varieties. One has the stratification [2] .
We will denote by: i) Π g = ∪ r≥2 A g,r = A g, [2] the locus of decomposable principally polarized abelian varieties; ii) A (ns) g the locus of classes of non-simple abelian varieties, i.e., of principally polarized abelian varieties of dimension g which are isogenous to a product of abelian varieties of dimension smaller that g; iii) A End =Z g the locus of classes of singular abelian varieties, i.e., of principally polarized abelian varieties whose endomorphism ring is larger than Z.
The locus Π g is reducible with irreducible components A g,p with p running through the partitions g = (i, g − i) of g for 1 ≤ i ≤ g/2 and we have codim Ag (A g,(i,g−i) ) = i(g − i). In contrast to this A End =Z g and A irreducible closed subsets of A g of codimension at least g − 1, the minimum codimension being achieved for families of abelian varieties that are isogenous to products of an elliptic curve with an abelian variety of dimension g − 1 (compare with [7]).
We recall a result from [20] and the main result from [12].
Theorem 7.5. For every integer r with 2 ≤ r ≤ g one has: Hence, for every integer r such that 2 ≤ r ≤ g, one has the stratification whereas for every integer k such that 0 ≤ k ≤ g − 2, one has the stratification

Lower bounds for the codimension of Andreotti-Mayer loci
The results in the previous sections give information about the Zariski tangent spaces to these loci and this will allow us to prove bounds on the dimension of the Andreotti-Mayer loci, which is our main objective in this paper.
We start with the results on tangent spaces. We need some notation.
Definition 8.1. Let ζ = (X, Θ X ) represent a point in N g,k,r . By L g,k,r (ζ) we denote the linear system of quadrics P(T ⊥ N g,k,r ,ζ ), where T N g,k,r ,ζ is the Zariski tangent space and where we view P(T A g,ζ ) as a space of quadrics as in Section 2. As usual, we may drop the index r if r = 2 and write L g,k (ζ) for L g,k,2 (ζ).
Notice that Definition 8.2. For ζ = (X, Θ X ) ∈ N g,k,r we denote by Sing (k,r) (Θ X ) the locally closed subset Moreover, we define Q (k,r) ζ to be the linear system of quadrics in P g−1 = P(T X,0 ) spanned by the union of all linear systems Q Propositions 4.1 and 6.3 imply the following basic tool for giving upper bounds on the dimension of the Andreotti-Mayer loci.
Proposition 8.3. Let N be an irreducible component of N g,k,r with its reduced structure. If ζ = (X, Θ X ) is a general point of N then the projectivized conormal space to N at ζ, viewed as a subspace of P(Sym 2 (T ∨ X,0 )), contains the linear system Q for all x ∈ Sing k,r (Θ X ). Since we work in characteristic 0 the map dπ is surjective on the tangent spaces for general points m ∈ M and π(m) ∈ N . Therefore the result follows from Propositions 4.1 and 6.3.
We need a couple of preliminary results. First we state a well-known fact, which can be proved easily by a dimension count.
Next we prove the following: If all polar quadrics of V coincide, then V is a hyperplane H counted with multiplicity d, and the polar quadrics coincide with 2H.
Proof. If d = 3 the assertion follows from general properties of duality (see [41], p. 215) or from an easy calculation. If d > 3, then the result, applied to the cubic polars of V , tells us that all these cubic polars are equal to 3H, where H is a fixed hyperplane. This immediately implies the assertion.
The next result has been announced in [8].
Theorem 8.6. Let g ≥ 4 and let N be an irreducible component of N g,k not contained in N g,k+1 . Then: Proof. By Theorem 7.5 and Remark 7.4, we may assume k < g − 2. By definition, there is some irreducible component M of π −1 Sg (N ) with dim(M ) = dim(N )+k which dominates N via π. We can take a general point (X, Θ X , z) ∈ M so that ζ = (X, Θ X ) is a general point in N . By Remark 7.4 we may assume X is simple. Let R be the unique k-dimensional component of π −1 |Sg (ζ) containing (X, Θ X , z). Its general point is of the form ξ = (X, Θ X , x) with x the general point of the unique k-dimensional component of Sing(Θ X ) containing z, and x has multiplicity r on Θ X . By abusing notation, we may still denote this component by R. Proposition 6.3 implies that the linear system of quadrics P(T ⊥ N,ζ ) contains all polar quadrics of T C ξ with ξ = (X, Θ X , x) ∈ R. Thus we have a rational map It is useful to remark that the quadric Q b,ξ is a cone with vertex containing the projectivized Zariski tangent space to R at x (see Remark 6.4).
Claim 8.7 (The finiteness property). For each b ∈ (P g−1 ) r−2 the map φ restricted to {b} × R has finite fibres.
Proof of the claim. Suppose the assertion is not true. Then there is an irreducible curve Z ⊆ R such that, for ξ = (X, Θ X , x) corresponding to the general point in Z, one has Q b,ξ = Q. Set Π = Vert(Q), which is a proper subspace of P g−1 .
Consider the Gauss map which associates to a smooth point of Z its projectivized tangent space. Then Proposition 6.3 implies that γ(ξ) ∈ R ⊆ Vert(Q ξ ) = Π. Thus γ(Z) is degenerate in P g−1 and this yields that X is non-simple, cf. [31]. This is a contradiction which proves the claim.
Claim 8.7 implies that the image of the map φ has dimension at least k, hence codim Ag (N g,k ) ≥ k + 1. To do better we need the following information. To prove part (ii) we now want to prove that dim(Q (r) ζ ) > k + 1. Remember that the image of φ has dimension at least k by Claim 8.7. If the image has dimension at least k + 1, then by Claim 8.8 it cannot be a projective space and therefore dim(Q (r) ζ ) > k + 1. So we can assume that the image has dimension k. Therefore each component of the fibre F Q over a general point Q in the image has dimension (g − 1)(r − 2). Consider now the projection of F Q to R. If the image is positive-dimensional then there is a curve Z in R such that the image of the Gauss map of Z is contained in the vertex of Q. Then X is non-simple, a contradiction (see the proof of Claim 8.7). Therefore the image of F Q on R is constant, equal to a point ξ, hence F Q = (P g−1 ) r−2 ×{ξ}. By Lemma 8.5 there is a hyperplane H ξ such that T C ξ = rH ξ , and Q = 2H ξ . Therefore we have a rational map sending ξ to H ξ . We notice that the image of φ is then equal to the 2-Veronese image if the image of ψ. By Claim 8.7, the map ψ has finite fibres. By an argument as in Claim 8.8, we see that the image of ψ does not contain a line, hence is not a linear space. Thus it spans a space of dimension s ≥ k +1. Then its 2-Veronese image, which is the image of φ spans a space of dimension at least 2s ≥ 2k + 2 > k + 2.
To prove part (iii), by part (ii), we can assume r = 2. It suffices to show that ζ . Suppose instead that dim(Q ζ ) = k + 1 and set Σ = φ(R). We have to distinguish two cases: (a) not every quadric in Q ζ is singular; (b) the general quadric in Q ζ is singular, of rank g − ρ < g. In case (a), consider the discriminant ∆ ⊂ Q ζ , i.e. the scheme of singular quadrics in Q ζ . This is a hypersurface of degree g, which, by Proposition 4.4, contains Σ with multiplicity at least k. Thus deg(Σ) ≤ g/k ≤ 3 and Σ contains some line, so that we have the corresponding pencil L of singular quadrics. By Claim 8.8, one arrives at a contradiction. Now we treat case (b). Let g − h be the rank of the general quadric in Σ. One By applying part (iii) of Segre's Theorem 21.2 to a general pencil contained in Q ζ , we deduce that (18) 3ρ ≤ g + 2s + 2.
Claim 8.9. One has s < r − k.
Proof. Suppose s ≥ ρ − k. If Π ′ is a general subspace of Π of dimension ρ − k, then its intersection with P(T R,x ), where x ∈ R is a general point, is not empty. Since ρ − k < g − k and X is simple, this is a contradiction (see [31], Lemma II.12).
By (18) and Claim 8.9 we deduce that ρ + 2k ≤ g and therefore 3k ≤ g, a contradiction. Suppose now ρ < h. Then part (iv) of Segre's Theorem 21.2 yields The right-hand-side is an increasing function of ρ, thus deg( By Lemma 8.4 the locus Σ contains a line, and we can conclude as in the proof of the non-degeneracy property 8.8.
The following corollary was proved independently by Debarre [9] and includes a basic result by Mumford [27].
Corollary 8.10. Let g ≥ 4. Then: is a general point of S, then either Θ X has isolated singularities or (X, Θ X ) is a product of an elliptic curve with a principally polarized abelian variety of dimension g − 1; then Θ X has isolated singularities. Moreover for every point x ∈ Sing(Θ X ), the quadric Q x is smooth and independent of x.
Proof. Take an irreducible component S of S g and let N be its image via the map π : Then by Theorem 8.6 the codimension of N in A g is at least k + 2. This implies that the codimension of S in X g is at least g + 2, which is impossible because S g is locally defined in X g by g + 1 equations. Then either k = 0 or k = g − 2.
In the former case S maps to an irreducible component N of N g,0 which is a proper subvariety of codimension 1 in A g . If k = g − 2, then by Theorem 7.5 the polarized abelian variety (X, Θ X ) is a product of principally polarized abelian varieties. The resulting abelian variety has a vanishing thetanull and so N is contained in the divisor of a modular form (a product of thetanulls, cf. e.g. [27], p. 370) which is a component of Finally, we show a basic property of S g .
Theorem 8.11. The locus S g is reduced.
Proof. The assertion is well-known for g ≤ 3, using the theory of curves. We may assume g ≥ 4.
Let S be an irreducible component of S g , let N = π(S) and k ≤ g − 2 the maximal integer such that N is contained in N g,k . As in the proof of Corollary 8.10, one has either k = 0 or k = g − 2. Assume first k = 0, and let ξ = (X, Θ X , x) ∈ S be a general point. We are going to prove that Sing(Θ X ) is reduced of dimension 0. First we prove that x has multiplicity 2 for Θ X . Suppose this is not the case and x has multiplicity r ≥ 3. Since N g,0 has codimension 1 all polar quadrics of T C ξ are the same quadric, say Q (see Proposition 6.3 and 8.3). By Lemma 8.5 the tangent cone T C ξ is a hyperplane H with multiplicity r. Again by Proposition 6.3 all the derivatives of Q with respect to points b ∈ H coincide with Q. By Proposition 5.3 the scheme D b,2 supported at x is contained in Sing(Θ X ). By taking into account Proposition 5.4 and repeating the same argument we see that this subscheme can be indefinitely extended to a 1-dimensional subscheme, containing x and contained in Sing(Θ X ). This implies that the corresponding component of N g,0 is contained in N g,1 , which is not possible since the codimension of N g,1 is at least 3. If x has multiplicity 2 the same argument shows that the quadric Q ξ is smooth. By Corollary 3.4, ξ is a smooth point of S g and this proves the assertion. Suppose now that k = g − 2. Then by Theorem 7.5 N is contained in the locus of products A g,(1,g−1) , and for dimension reasons it is equal to it and then the result follows from a local analysis with theta functions.
then Θ X has finitely many double points with the same tangent cone which is a smooth quadric.

A conjecture
As shown in [8], part (i) of Theorem 8.6 is sharp for k = 1 and g = 4 and 5. However, as indicated in [8], it is never sharp for k = 1 and g ≥ 6, or for k ≥ 2. In [8] we made the following conjecture, which is somehow a natural completion of Andreotti-Mayer's viewpoint in [2] on the Schottky problem. Recall the Torelli morphism t g : M g → A g which maps the isomorphism class of a curve C to the isomorphism class of its principally polarized Jacobian (J(C), Θ C ). As a map of orbifolds it is of degree 2 for g ≥ 3 since the general abelian variety has an automorphism group of order 2 and the general curve one of order 1. We denote by J g the jacobian locus in A g , i.e., the Zariski closure of t g (M g ) in A g and by H g the hyperelliptic locus in A g , that is, the Zariski closure in A g of t g (H g ), where H g is the closed subset of M g consisting of the isomorphism classes of the hyperelliptic curves. By Torelli's theorem we have dim(J g ) = 3g − 3 and dim(H g ) = 2g − 1 for g ≥ 2.
Moreover, equality holds if and only if one of the following happens: (i) g = k + 3 and N = H g ; (ii) g = k + 4 and N = J g .
By work of Beauville [4] and Debarre [11] the conjecture is true for g = 4 and g = 5. Debarre [10,11] gave examples of components of N g,k for which the bound in Conjecture 9.1 for the codimension in A g fails, but they are contained in A End =Z g , since the corresponding abelian varieties are isogenous to products. Our main objective in this paper will be to prove the conjecture for k = 1.
Remark 9.2. The question about the dimension of the Andreotti-Mayer loci is related to the one about the loci S g,k introduced in §3. Note that M g,k = π * (N g,k ) is a subscheme of S g,k . Let N be an irreducible component of N g,k not contained in N g,k+1 , let M be the irreducible component of M g,k dominating N and let Z be an irreducible component of S g,k containing M . We now give a heuristic argument. Recalling Proposition 4.3, we can consider Z to be Since M is contained in Z, one has also codim Sg (M ) ≥ k+1 2 and therefore codim Ag (N ) ≥ k+2 2 , which is the first assertion in Conjecture 9.1. In this setting, the equality holds if and only if Z is well-behaved and M = Z. On the other hand, since M is got from Z by imposing further restrictions, one could expect that M is, in general, strictly contained in Z and therefore that codim . This is clearly the case if [resp. c < g−1 2 ] in A g , whose general point corresponds to a principally polarized abelian variety (X, Θ X ) with Sing(Θ X ) containing a subscheme isomorphic to Spec One might be tempted to believe that an affirmative answer to the first question implies a negative answer to the second. This is not the case. Indeed H 4 is contained in the locus of Jacobians of curves with an effective even theta characteristic. In this case S 4,1 has two well-behaved irreducible components of dimension 8, one dominating H 4 with fibres of dimension 1, the other one dominating J 4 ∩ θ 4,0 , where θ 4,0 is the theta-null locus, see §19 and §10 below and [16]. These two components intersect along a 7-dimensional locus in S 4 which dominates H 4 . Note that it is not always the case that a component of S g,k is well-behaved. For example, there is an irreducible component of S g,g−2 , the one dominating A g,(1,g−1) , which is also an irreducible component of S g .
10. An Example for Genus g = 4 In this section we illustrate the fact that the quadrics associated to the singularities of the theta divisor may provide more information than obtained above. The locus S 4 in the universal family X 4 consists of three irreducible components: i) one dominating θ 0,4 , the locus of abelian varieties with a vanishing theta-null; we call it A; ii) one dominating the Jacobian locus J 4 ; we call it B; iii) one dominating A 4, (3,1) , the locus of products of an elliptic curve with a principally polarized abelian variety of dimension 3. The components A and B have codimension 5 in X 4 and they intersect along a locus C of codimension 6. The image of C in A 4 is the locus of Jacobians with a half-canonical g 1 3 . The quadric associated to the singular point of order 2 of X in Θ X has corank 1 at a general point. We refer to [16] for a characterization of the intersection θ 0,4 ∩ J 4 . Proof. The scheme Sing(Θ X ) contains a scheme Spec(C[ǫ]) at x, say corresponding to the tangent vector D 1 ∈ ker q x , cf. Section 5. In order that ξ be a singular point of S 4 , the quadrics q and q 1 associated to x and D 1 (cf. equation (6) and Proposition 3.3) must be linearly dependent. By Proposition 5.3 this implies that Sing(Θ X ) contains a scheme of length 3 at x. Proof. Let ξ = (X, Θ X , x) be a general point of C so that the singular point x of Θ X defines a quadric q x of rank 3. Set η = (X, Θ X ) ∈ A 4 . From the equations (3) and (6), one deduces that the tangent cone to N 4,0 at η is supported on q ⊥ x . Suppose, as we may by applying a suitable translation, that x is the origin 0 in X. Then θ 0,4 is locally defined by the equation θ(τ, 0) in A 4 (see [16], §2). Hence θ 0,4 is smooth at η with tangent space q ⊥ x . Now the jacobian locus J 4 is also smooth at η by the injectivity of the differential of Torelli's morphism at non-hyperelliptic curves (see [15]) and the assertion follows.

The Gauss map and tangencies of theta divisors
In this section we shall study the situation where a number of translates of the theta divisor of a principally polarized abelian variety are 'tangentially degenerate,' that is, are smooth but with linearly dependent tangent spaces or singular at prescribed points. Let (B, Ξ) be a polarized abelian variety of dimension g, where Ξ is an effective divisor on B. As usual we let B = C g /Λ, with Λ a 2g-dimensional lattice, and p : C g → B be the projection. So we have coordinates z = (z 1 , . . . , z g ) in C g and therefore on B, and we can keep part of the conventions and the notation used so far. Let ξ = ξ(z) be the Riemann theta function whose divisor on C g descends to Ξ via p. If Ξ is reduced, then the Gauss map of (B, Ξ) is the morphism is the tangent space to Ξ at ξ translated to the origin. If Ξ b = t b (Ξ) is the translate of Ξ by the point b ∈ B defined by the equation ξ(z − b) = 0 then for x ∈ Ξ − Sing(Ξ) the origin is a smooth point for Ξ −x and γ(x) = P(T Ξ−x,0 ). As usual we have natural homogeneous coordinates (z 1 : . . . : z g ) in P(T B,0 ) = P g−1 and therefore dual coordinates in the dual projective space P(T ∨ B,0 ) = P g−1 ∨ . Then the expression of γ in coordinates equals γ(p(z)) = (∂ 1 ξ(z) : . . . : The following lemma is well-known.
Lemma 11.1. Let (B, Ξ) be a simple abelian variety of dimension g and Ξ an irreducible effective divisor on B. Then the map γ Ξ has finite fibres. Moreover, for a smooth point x ∈ Ξ there are only finitely many b ∈ B such that Ξ b is smooth at x and tangent to Ξ there.
Proof. Suppose γ Ξ does not have finite fibres. Then there is an irreducible curve C of positive geometric genus contained in the smooth locus of Ξ which is contracted by γ Ξ to a point of P g−1 ∨ corresponding to a hyperplane Π ⊂ P g−1 .
Then the image of the Gauss map γ C of C lies in Π. This implies that C does not generate B and contradicts the fact that B is simple. As to the second statement, if it does not hold there exists an irreducible curve C such that for all b ∈ C the divisor Ξ b is smooth at x and tangent to Ξ. Then the curve C ′ = {x − b : b ∈ C} is contained in Ξ and contracted by γ Ξ . This contradicts the fact that γ Ξ has finite fibres as we just proved.
. . . Proof. Let us first assume h = 1 and denote a component of T h (B, Ξ) by T . By composing p 1 with the Gauss map γ = γ Ξ , we obtain a rational map φ : T P(T ∨ B,0 ). We shall prove that φ has finite fibres. Let v be a point in the image of φ coming from a point in the open subset as in Definition 11.2, (ii). By Lemma 11.1 there are only finitely many smooth points z 1 , . . . , z a ∈ Ξ such that γ(z i ) = v for i = 1, . . . , a. For each 1 ≤ i ≤ a we consider the theta divisor defined by ξ(z i − u) = 0. Again by Lemma 11.1 there are only finitely many points u i1 , . . . , u iℓi in it such that (z i , u ij ) may give rise to a point on T . So φ has finite fibres. Thus it is dominant and T has dimension g − 1. Now we prove the assertion by induction on h. Consider the projection q : T → B × B h−1 by forgetting the last factor B. If the image T ′ of T is contained in T h−1 (B, Ξ), then it is contained in a regular component of T h−1 (B, Ξ), hence by induction the codimension of T is at least g + 1, while we know from the equations that it is at most g + 1, and the assertion follows.
be the universal bundle over the Grassmannian of (h − 1)-planes in P g−1 .
We will also consider the closed subscheme T 0 . . .
Finally we will consider the closed subset T h (X, B) which is the union of In case (a) we say that the divisors Ξ, Ξ u1 , . . . , Ξ u h , all passing through z with multiplicity one, are tangentially degenerate at z.
In case h = 1 we have only two divisors which are just tangent at z. We will drop the index h if h = 1. Thus, N 0 (B, Ξ) is the set of all u such that Ξ and Ξ u are either tangent, or both singular, somewhere. Note that, if Ξ is symmetric, i.e., if ξ(z) is even, as happens for the Riemann theta function, then N 0 (B, Ξ) contains the divisorial component 2Ξ := {2ξ : ξ ∈ Ξ}, since γ(ξ) = γ(−ξ) for all smooth points ξ ∈ Ξ. One has the following result by Mumford (see [27], Proposition 3.2).
As we will see later, the following result is related to Conjecture 9.1. We will prove now a more precise result in the spirit of Proposition 11.6. Proof. We may assume (B, Ξ) ∈ N g,k , otherwise there is nothing to prove. We may also assume h ≥ 1 is minimal under the hypothesis that N k,h (B, Ξ) has positive dimension. Let (u 1 , . . . , u h ) be a point in N k,h (B, Ξ), so that Ξ 0 := Ξ and Ξ i := Ξ ui , i = 1, . . . , h, are tangentially degenerate along a non-empty subset V 0 of an irreducible subvariety V of B of dimension k such that Ξ i for i = 0, . . . , h are smooth at all points v ∈ V 0 . For j = 0, . . . , h the element s (i) j := ∂ i ξ(z − u j ) (with the convention that u 0 = 0) is a section of O(Ξ j ) when restricted to V since Ξ j contains V . We know that for given j not all s (i) j are identically zero on V 0 . Our assumptions on the tangential degenerateness and minimality tell us that there exist non-zero rational functions a j such that we have a non-trivial relation (20) (20) in different patches which trivialize the involved line bundles, and comparing them, we see that, if the transition functions are not all proportional, then we can shorten the original relation by subtracting two of them. This would contradict the minimality assumption. Therefore, we have that all of the divisors f * (Ξ − Ξ uj ) |V with j = 1, . . . h are linearly equivalent. Since u = (u 1 , . . . , u h ) varies in a subvariety of positive dimension this implies that the map Pic 0 (B) → Pic(W ) has a positive-dimensional kernel, and this is impossible since B is simple. So there exists an index j such that the function a j has poles on a divisor Z j of W . Note now that a point which is non-singular for all the divisors Ξ j , j = 1, . . . , h, is certainly not a pole for the functions a j . Therefore, for each j = 1, . . . , h, there an ℓ depending of j such that Z j is contained in the divisorial part of the scheme f * (Sing(Ξ ℓ )). Moreover, since Z j moves in a linear system on W , it cannot be contracted by the birational morphism f . This proves that Ξ ℓ is singular along a variety of dimension k − 1 contained in V , which proves the assertion. Remark 12.3. If we have two divisors Ξ 0 , Ξ 1 which are tangentially degenerate along an irreducible k-dimensional variety V whose general point is smooth for both Ξ 0 , Ξ 1 , then Ξ 0 , Ξ 1 are both singular along some (k−1)-dimensional variety contained in V . This can be easily proved by looking at the relation (20) in this case, and noting that the polar divisor Z j is contained in f * (Sing(Ξ j )), j = 0, 1.
Remark 12.4. Suppose (B, Ξ) / ∈ N g,0 . Then N 0 (B, Ξ) is described by all differences of pairs of points of Ξ having the same Gauss image. Suppose (B, Ξ) ∈ N g,0 − N g,1 and assume {x, −x} = Sing(Ξ) have multiplicity 2 and the quadric Q x = Q −x is smooth. It may or may not be the case that b = −2x ∈ N 1 (B, Ξ). In any case, we claim that N 1 (B, Ξ)− {−2x} is contained in the set of all differences of points in γ −1 Ξ (Q * x ) with x. Let us give a sketch of this assertion, which provides, in this case, a different argument for the proof of Proposition 11.6. If b ∈ N 1 (B, Ξ) − {−2x}, there is a curve C ⊂ Ξ such that for t ∈ C general γ Ξ (t) = γ Ξ (t + b). Along C the divisors Ξ and Ξ b are tangent, hence the curve contains x (see Remark 12.3). Note that the curve C is smooth at x. Indeed, locally at x, the divisor Ξ is a quadric cone of rank g − 1 in C g with vertex x, whereas Ξ b is a hyperplane through x, and they can only be tangent along a line.
Thus it makes sense to consider the image of x for γ C , which is a point on Q x . The point x + b ∈ C + b ⊂ Ξ is smooth for Ξ and γ(x + b) is clearly tangent to Q x at γ C (x). 13. On N g−2 and N g−3 for Jacobians The following result shows that the bound in Proposition 11.6 is sharp in the case h = 1. Recall that a curve C is called bielliptic if it is a double cover of an elliptic curve.
Proposition 13.1. Let C be a smooth, irreducible projective curve of genus g and let (J = J(C), Θ C ) be its principally polarized Jacobian. Then one has: Proof. We begin with (i). We assume that C is not hyperelliptic; the hyperelliptic case is similar and can be left to the reader. We may identify C with its canonical image in P g−1 . Moreover, we identify J with Pic g−1 (C) and Θ C ⊂ Pic g−1 (C) with the set of effective divisor classes of degree g−1. Then the Gauss map γ C := γ ΘC can be geometrically described as the map which sends the class of an effective divisor D of degree g − 1 such that h 0 (C, O C (D)) = 1 to the hyperplane in P g−1 spanned by D (see [15], p. 360). Take two distinct points p 1 , p 2 on C. Then |ω C (−p 1 − p 2 )| is a linear series of degree 2g − 4 and dimension g − 3. For i = 1, 2 we let V i be the subvariety of Θ C which is the Zariski closure of the set of all divisor classes of the type E + p i , where E is a divisor of degree g − 2 contained in some divisor of |ω C (−p 1 − p 2 )|. Clearly dim(V i ) = g − 3, hence V i is not contained in Sing(Θ C ) which is of dimension g −4. If u denotes the divisor class p 2 −p 1 then x → x+u defines an isomorphism V 1 ∼ = V 2 . Moreover, for x in an non-empty open subset of V 1 we have γ C (x) = γ C (x + u). This proves (i). Conversely, assume there is a point u ∈ Pic 0 (C) − {0}, and a pair of irreducible subvarieties V 1 , V 2 of Θ C of dimension g − 3 such that x → x + u gives a birational map from V 1 to V 2 . For x in a non-empty open subset U ⊂ V 1 we have γ C (x) = γ C (x + u). If D and D ′ are the effective divisors of degree g − 1 on C corresponding to x and x + u and if E is the greatest common divisor of D and D ′ then by the geometric interpretation of the Gauss map γ C there is Consider the linear series |K C − u|, which is a g g−2 2g−2 . If this linear series has a base point q, then there is a point p such that K C − u − q ≡ K C − p, i.e. D ′ − D ≡ u ≡ p − q, proving (ii). So we may assume that |K C − u| has no base point. If C is not bielliptic, then |K C − u| determines a birational map of C to a curve in P g−2 . On the other hand it contains the (g − 3)-dimensional family of divisors of the form (2D −E)+F , which are singular along the divisor D − E. This is only possible if deg(D − E) = 1, i.e., u ≡ D ′ − D ≡ p − q, with p, q ∈ C. But in this case |K C − u| has the base point q, a contradiction. This proves (ii). Assume now C is hyperelliptic. Let p 1 + p 2 be an effective divisor of the g 1 2 on C with p 1 = p 2 . Then |ω C (−p 1 − p 2 )| is a linear series of degree 2g − 4 and dimension g − 2. For i = 1, 2 we let V i be the subvariety of Θ C which is the Zariski closure of the set of all divisors classes of the type E + p i , where E is a divisor of degree g − 2 contained in a divisor of |ω C (−p 1 − p 2 )|. The variety V i has dimension g − 2 and is not contained in Sing(Θ C ), which is of dimension g − 3. The translation over u induces an isomorphism V 1 ∼ = V 2 and for x in a non-empty subset U of V 1 we have γ C (x) = γ C (x + u). Hence the left-hand-side in (iii) is contained in N g−2 (J, Θ C ). Finally, assume there is a point u ∈ Pic 0 (C) − {0}, and a pair of irreducible subvarieties V , V ′ of Θ C of dimension g − 2 such that translation by u gives a rational map V V ′ with γ C (x) = γ C (x + u) on a non-empty open subset of V . Let D, D ′ be the effective divisors of degree g − 1 on C corresponding to x and x + u. As in the proof of part (ii), let E be the greatest common divisor of D and D ′ . In the present situation the linear series |K C − u| of dimension g − 2 contains the (g − 2)-dimensional family of divisors of the form 2D − E, which are singular along the divisor D − E. This means that 2(D − E) is in the base locus of |K C − u|. This is only possible if D = E + q for some point q ∈ C, and K C − u − 2q ≡ (g − 2)(p + q), where p is conjugated to q under the hyperelliptic involution. In conclusion, we have u ≡ p − q and the equality in (iii) follows.
Remark 13.2. The hypothesis that C is not bielliptic is essential in (ii) of Proposition 13.1. Let in fact C be a non-hyperelliptic bielliptic curve which is canonically embedded in P g−1 . Let f : C → E be the bielliptic covering. One where B is the branch divisor of f . Let u ∈ Pic 0 (E) − {0} be a general point, which we can consider as a nontrivial element in Pic 0 (C) via the inclusion f * : Pic 0 (E) → Pic 0 (C) Note that f is ramified, hence f * is injective. We want to show that u ∈ N g−3 (J, Θ C ), proving that equality does not hold in (i) in this case. The canonical image of C is contained in a cone X with vertex a point v ∈ P g−1 over the curve E embedded in P g−2 as a curve of degree g − 1 via the linear system |ξ|. Let us consider the subvariety W of |ξ| consisting of all divisors M ∈ |ξ| such that there is a subdivisor p + q of M with p, q ∈ E and p − q ≡ u. It is easily seen that W is irreducible of dimension g − 3.
Notice that for M general in W one has M = p + q + N , with N effective of degree g − 3. Therefore we may write where F, F ′ are disjoint, effective divisors of degree g − 3 which are exchanged by the bielliptic involution. We let V be the (g − 3)-dimensional subvariety of Θ C described by all classes of divisors D of degree g − 1 on C of the form D = f * (p) + F , as M varies in W . Then D + u ≡ D ′ := f * (q) + F and D and D ′ span the same hyperplane through v in P g−1 . Therefore, if x ∈ V is the point corresponding to D, one has γ C (x) = γ C (x + u). This proves that u ∈ N g−3 (J, Θ C ).

A boundary version of the Conjecture
We will now formulate a conjecture. As we will see later, it can be considered as a boundary version of Conjecture 9.1 (see also Proposition 11.6). (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.
One implication in this conjecture holds by Proposition 13.1. Note that the conjecture would give an answer for simple abelian varieties to the Schottky problem that asks for a characterization of Jacobian varieties among all principally polarized abelian varieties. For related interesting questions, see [30].

Semi-abelian Varieties of Torus Rank One
Let (B, Ξ) be a principally polarized abelian variety of dimension g − 1. The polarization Ξ gives rise to the isomorphism and we shall identify B andB via this isomorphism. Thus an element b ∈ B ∼ = B determines a line bundle L = L b = O B (Ξ − Ξ b ) with trivial first Chern class. We can associate to L a semi-abelian variety X = X B,b , namely the G mbundle over B defined by L which is an algebraic group since it coincides with the theta group G b := G(L) of L (cf. [25], p. 221). This gives the well-known equivalence betweenB and Ext(B, G m ), the group of extension classes of B with G m in the category of algebraic groups (see [35], p. 184). Both the line bundle L and the G m -bundle X determine a P 1 -bundle P = P(L ⊕ O B ) over B with projection π : P → B and two sections over B, say s 0 and s ∞ given by the projections L ⊕ O B → L and L ⊕ O B → O B . If we set P 0 = s 0 (B) and P ∞ = s ∞ (B) then we can identify P − P 0 − P ∞ with X. By [17], Proposition 2.6 on page 371 we have O P (1) ∼ = O P (P 0 ). We can complete X by considering the non-normal variety X = X B,b obtained by glueing P 0 and P ∞ by a translation over b ∈ B ∼ =B. On P we have the linear equivalence P 0 − P ∞ ≡ π −1 (Ξ− Ξ b ). We set E := P ∞ + π −1 (Ξ) and put M P = O P (E). This line bundle restricts to O B (Ξ) on P 0 and to O B (Ξ b ) on P ∞ , and thus descends to a line bundle M = M X on X. We have π * (O P (E)) = O B (Ξ) ⊕ O B (Ξ b ) and H 0 (P, M P ) is generated by two sections with divisors P ∞ + π −1 (Ξ) and P 0 + π −1 (Ξ b ). One concludes that H 0 (X, M ) corresponds to the sections of M P such that translation over b carries its restriction to P 0 to the restriction to P ∞ . It follows that h 0 (X, M ) = 1 with effective divisor Ξ, which is called the generalized theta divisor on X. Analytically we can describe a section of OX (Ξ) on the universal cover C×C g−1 by a function ξ(τ, z) + u ξ(τ, z − ω), where ω ∈ C g−1 represents b ∈ B = C g−1 /Z g−1 + τ Z g−1 , ξ(τ, z) is Riemann's theta function for B and u = exp(2πiζ) is the coordinate on C * . This is called the generalized theta function of X.
Let D be the Weil divisor on X that is the image of P 0 (or, what is the same, of P ∞ ). We consider a locally free subsheaf T vert of the tangent sheaf to X, namely the dual of the sheaf Ω 1 (log D) of rank g. If d ∈ D is a point such that on the normalization P near the two preimages z 1 , . . . , z g−1 , u and z 1 +b, . . . , z g−1 +b, v are local coordinates such that u = 0 defines P 0 (resp. v = 0 defines P ∞ ) with uv = 1 then T vert is generated by ∂/∂z 1 , . . . , ∂/∂z g−1 , u∂/∂u − v∂/∂v.
The points in Sing vert (Ξ) are of two sorts depending on whether they lie on the double locus D of X or not. The singular points of Sing(Ξ) on X = X − D are the points (z, u), with u = 0, which are zeros of ξ(τ, z) and ξ(τ, z − ω) and such that γ(τ, z) = −uγ(τ, z − ω). That is, geometrically, these correspond under the projection on B to the points on B where Ξ and Ξ b are tangent to each other. To describe the singular points of Sing(Ξ) on D, we consider the composition φ : where ν is the normalisation. Then we have φ −1 (Sing vert (Ξ)) = Sing(Ξ) and the same if we identify B with P ∞ . Points of Sing vert (Ξ) determine again quadrics in P g−1 as follows. Note that the projective space P(T X,0 ) contains a point P b corresponding to the tangent space T Gm ⊂ T X of the algebraic torus G m at the origin. Recall that we write γ(τ, z) for the row vector γ(τ, z) = (∂ 1 ξ, . . . , ∂ g−1 ξ)(τ, z).
Then a singular point determines a, possibly indeterminate, quadric defined by the matrix ). Note that we have γ(τ, z) = −uγ(τ, z − ω). The quadric passes through the point P b . For a point on D the quadric is a cone with vertex P b over a quadric in P g−2 given by M .

Standard Compactifications of Semi-Abelian Varieties
Let (B, Ξ) be a principally polarized abelian variety. We assume now that dim(B) = g − r with r ≥ 1 and extend the considerations of the previous section. The extensions of B by G r m are parametrized by Ext 1 (B, G r m ) ∼ =B r . To a point (b 1 , . . . , b r ) ∈B r one associates the G r m -extension X = X b obtained as the fibre product of theta groups G b1 × B · · · × B G br . One of the type of degenerations of abelian varieties that we shall encounter are special compactifications of semi-abelian varieties. We shall call them standard compactifications of torus rank r. Let b = (b 1 , . . . , b r ) ∈B r . The algebraic group X = X b sits in a P 1 × B · · · × B P 1 -bundle π : P → B that is obtained as the fibre product over B of the P 1 -bundles P bi = P(L bi ⊕O B ). The complement P − X is a union of 2r divisors is given by taking 0 (resp. ∞) in the i-th fibre coordinate, with projections π i,0 , π i,∞ to B. We now define a non-normal variety obtained from P by glueing Π (i) 0 with Π (i) ∞ for i = 1, . . . , r. This identification depends on a r × r-matrix T = (t ij ) with entries from G m such that t ii = 1 and t ij = t −1 ji . Let s (i) 0 : B → P bi (resp. s (i) ∞ ) be the zero-section (infinity section) of P bi . We glue the point We denote the resulting variety by X. It depends on the parameters b ∈B r and t ∈ Mat(r × r, G m ). We have the linear equivalences Π Thus, by the definition of the glueing, M descends to a line bundle M := M X on X. We have Hence we have h 0 (P, M ) = 2 r . As in the preceding section one sees that only a 1-dimensional space of sections descends to sections of M on X. In terms of coordinates (ζ 1 , . . . , ζ r , z 1 , . . . , z g−r ) on the universal cover of X, where (z 1 , . . . , z g−r ) ∈ C g−r are coordinates on the universal cover of B, a non-zero section of M is given by I⊆{1,...,r} where I runs through the subsets of {1, . . . , r}, u I = i∈I u i with u i = exp(2πζ i ), t I = i,j∈I,i<j t ij , b I = i∈I b i and ω I ∈ C g−r represents b I ∈ B. This is the generalized theta function of X, whose zero locus is the generalized theta divisor Ξ of X.
Next we look at the singular points of Ξ. All points in Ξ ∩D are singular points of Ξ. However, just as in the rank one case in the preceding section we will in general disregard these singularities of Ξ, and we will only look at the so-called vertical singularities, which we are going to define now (cf. [27], §2). The locally free subsheaf T vert of rank g of the tangent sheaf T X is the dual of Ω 1 (log D). Its pull back to P is generated, in the (u, z)-coordinates, by the differential operators u i ∂/∂u i − v i ∂/∂v i with u i v i = 1 for i = 1, . . . , r and ∂ j = ∂/∂z j with j = 1, . . . , g − r. We interpret local sections of T vert as derivations as above and define the scheme Sing vert (Ξ) of vertical singular points of Ξ as the subscheme of Ξ defined by the equations (21) with f = 0 a local equation of Ξ for all local sections ∂ ∈ T vert . This is independent of the choice of a local equation.
Proof. The compactification X is a stratified space and the (closed) strata are (standard) compactifications of semi-abelian extensions of (B, Ξ) of torus rank s with 0 ≤ s ≤ r. In view of the relations I u I t I ∂ i ξ(z − ω I ) = 0 the vertical singularities of Ξ correspond to points where 2 s translates of Ξ are tangentially degenerate. Therefore we have dim (N 1,h (B, Ξ)) ≥ 1 with h = 2 s . By Lemma 12.1 it follows that (B, Ξ) ∈ N g−r,0 .

Semi-abelian varieties of torus rank two
In the compactification of the moduli space of principally polarized abelian varieties of dimension g we shall encounter two types of degenerations of torus rank 2. The first of these is a standard compactification introduced above and its normalization is a P 1 × P 1 -bundle over a principally polarized abelian variety of dimension g − 2. For such a standard compactification the equations for Sing vert (Ξ) are given in terms of the (u, z)-coordinates by the system (we write t instead of t 1,2 ; note that t = 0) From this and the analogous equations in the v-coordinates (with u i v i = 1) we see that the vertical singular points of Ξ are essentially of three types: i.e. if and only if z and z − b are in Ξ and γ Ξ (z) = γ Ξ (z − b).
(iii) A point x / ∈ D, (i.e. in the (u, z) coordinates one has u 1 = 0 = u 2 ) is a vertical singularity if and only if z is a singular point of the divisor H ∈ |2Ξ b1+b2 | defined by the equation By the way, this occurs even in case (ii) above. Note also that, by the above equations, the existence of a vertical singularity implies that the theta divisors Ξ, Ξ b1 , Ξ b2 and Ξ b1+b2 are tangentially degenerate at some point x of B, i.e., z ∈ N 0,3 (B, Ξ). We call of type (i), (ii) or (iii) the singular points of Ξ according to whether cases (i), (ii) or case (iii) occur.
Remark 17.1. A point x in Sing vert (Ξ) again determines a quadric Q x in P g−1 .
It is useful to remark that: (a) in case (i) the quadric Q x is a cone with vertex the line L b := P b1 , P b2 given by the tangent space to the G m -part over the quadric Q z in P g−3 which corresponds to the singular point z of Ξ; (b) in case (ii), say we are at a point with u 1 = 0, u 2 = 0. Then Q x is a cone with vertex P b1 over the quadric in the hyperplane u 1 = 0 with matrix with γ(τ, z) = (∂ 1 ξ, . . . , ∂ g−2 ξ)(τ, z) and the matrix M is given by In §15 we saw that all rank 1 compactifications of G m -extensions of a principally polarized abelian variety B form a compact familyB. This is no longer the case in the higher rank case. This is where semi-abelian varieties of non-standard type come into the picture. This will depend on choices. It is good to see this in some detail in the rank 2 case. Given a principally polarized abelian variety (B, Ξ) of dimension g − 2, all standard rank 2 compactifications of (B, Ξ) are of the form (X, Ξ) with X = X B,b,t with b = (b 1 , b 2 ) ∈ B × B and t ∈ C * . Thus the parameter space may be identified (up to dividing by automorphisms) with the total space of the Poincaré bundle P → B × B minus the 0-section P 0 . It is then natural to compactify this by looking at the associated P 1 -bundle and by glueing on it the 0-section P 0 with the infinity section P ∞ . This in fact works and it is explained in [26], §7, and in [29]. We describe next the new objects that arise. We denote by L i the line bundle associated to b i , for i = 1, 2. We consider again the P 1 × P 1 -bundle P on B as in §16 and in the glueing operation described in §16, we let t = t 12 tend to 0 (or equivalently to ∞). Letting t → ∞, one contracts Π 1,0 and Π 2,0 to the section A = Π 1,∞ ∩ Π 2,∞ ∼ = B, and Π 1,∞ and Π 2,∞ to the section ∆ = Π 1,0 ∩ Π 2,0 ∼ = B. In order to properly describe the glueing process, we have first to blow up the two sections A and ∆ in P . Let us do that. Let w :P → P be the blow-up, on which we have the following divisors: α is the exceptional divisor over A and δ is the exceptional divisor over ∆; β, γ, ǫ, ζ are the proper transforms onP of Π 1,∞ , Π 2,0 , Π 1,0 , Π 2,∞ , respectively. We will abuse notation and denote by the same letters the restrictions of these divisors on the general fibre Φ ofP over B, which is a P 1 × P 1 blown up at two points, hence a P 2 blown up at three points. Note that α, β, γ, δ, ǫ, ζ are P 1 -bundles over B and one has At this point one could be tempted to suitably glue α with γ and ǫ and δ with β and ζ. This however, as (23) shows, does not work. The right construction is instead the following. One considers two P 2 -bundles φ i : P ♯ i → B, i = 1, 2, associated to the vector bundles L 1 ⊕ L 2 and L ∨ 1 ⊕ L ∨ 2 on B, i.e., . There are three relevant P 1 -subbundles of the bundles P ♯ i , i = 1, 2, namely (23) and (24) show, we can glue P with P ♯ 1 and P ♯ 2 in such a way that α and δ are respectively glued toᾱ andδ; ǫ is glued toǭ and β toβ with a shift by −b 1 , and ζ toζ and γ toγ with a shift by −b 2 . The resulting variety is denoted by X = X B,b . As usual, we will denote by D its singular locus. OnP we have the line bundlẽ where we write L instead of w * (π * (L)) for a line bundle, or divisor, L on B.
in the (u, z)-coordinates. Note that (27) is just obtained from the generalized theta function in 16 by letting t = t 12 tend to 0.
(ii) Consider a point x ∈ D which is the image via ϕ of a point in Π i,0 but not of a point in Π 3−i,0 , i.e. in the (u, z) coordinates one has u i = 0, u 3−i = 0, for an i = 1, 2. If i = 1, this is a vertical singular point of Ξ if and only if ξ(τ, z) = 0, ξ(τ, z − ω 2 ) =0, i.e., if and only if z and z − b 2 are in Ξ and γ Ξ (z) = γ Ξ (z − b 2 ). Thus points of this type correspond to points in N 0 (B, Ξ).
Remark 17.2. A point x in Sing vert (Ξ) determines a quadric Q x in P g−1 . Remark 17.1 is still valid here.
A variant of this second type of rank-2 degeneration is obtained as follows. Given a G 2 m -extension X = X b of B determined by a point b = (b 1 , b 2 ) ∈B 2 we consider two P 2 -bundles P and P ′ over B: and where we write as before L i for L bi . We can glue these over the common P 1subbundle P(L 1 ⊕ L 2 ). Then we glue the P 1 -subbundle P(O B ⊕ L 1 ) of P with the P 1 -subbundle P(L 2 ⊕ (L 1 ⊗ L 2 )) of P ′ via a shift over b 2 . Similarly, we glue the P 1 -subbundle P(O B ⊕ L 2 ) of P with the P 1 -subbundle P(L 1 ⊕ (L 1 ⊗ L 2 )) of P ′ via a shift over b 1 . In this way we obtain a non-normal variety over B.
Both P and P ′ come with a relatively ample bundle O P (1) and O P ′ (1). On P we have the linear equivalences with Π i = P(O B ⊕ L i ) for i = 1, 2 and Π 3 = P(L 1 ⊕ L 2 ). We let M be the line bundle O(Π 3 + π * (Ξ b1+b2 )) on P and M ′ the line bundle O(Π ′ 3 + π * (Ξ b1+b2 )) on P ′ , where Π ′ 3 is the bundle P(L 1 ⊕ L 2 ). This descends to a line bundle M on X. This line bundle has a 1-dimensional space of sections generated by in suitable affine coordinates (u 1 , u 2 ) on P 2 . Again the vertical singular points of Ξ are essentially of three types: is a vertical singularity if it corresponds to a singularity on Ξ, to a singularity on Ξ b1 on Π 1 ∩ Π 3 and a singularity on Ξ b2 on Π 2 ∩ Π 3 ; (ii) A point x ∈ D which is the image via ϕ of a point on one Π 3 but not of a point in Π 1 or Π 2 is a vertical singular point of Ξ if and only if x ∈ Ξ b1 ∩Ξ b2 and γ Ξ b 1 (x) = γ Ξ b 2 (x). Thus points of this type correspond to points in N 0 (B, Ξ). Something similar happens for the points on exactly one of Π 1 or Π 2 .
(iii) A point x / ∈ D is a vertical singularity if x ∈ Ξ ∩ Ξ b1 ∩ Ξ b2 and Ξ, Ξ b1 , Ξ b2 are tangentially degenerate at x. In other words points of this type correspond to points in N 0,2 (B, Ξ). We see that the compactification depends on a choice, but in both cases we can deal explicitly with the singularities of the theta divisors.

Compactification of A g
In order to study the Andreotti-Mayer loci we need to compactify A g . The moduli space A g admits a 'minimal' compactification, the Satake compactification constructed by Satake and Baily-Borel in characteristic 0 and by Faltings-Chai over the integers (see [32], [13]). This compactification A * g is an orbifold or stack which admits a stratification This compactification is highly singular for g ≥ 2. Smooth compactifications can be constructed by the theory developed by Mumford and his co-workers in characteristic 0 and by Faltings-Chai in general. These compactifications depend on combinatorial data. We shall use the Voronoi compactificationÃ g =Ã Vor g as described by Namikawa, Nakamura and Alexeev (see [29,28,5]). This compactification is a smooth orbifold with a natural map q :Ã g → A * g . It has the stratification induced by that of A * g : the stratum A (r) g = q −1 (A g−r ) is called the stratum of quasi-abelian varieties of torus rank r. In §20 we also shall use another compactification, the perfect cone compactification, cf. [29,34]. It also has a map to the Satake compactification denoted by q :Ã pc g → A * g ; sometimes we shall denoteÃ pc g simply byÃ g in order to avoid introducing more notation. It has the following properties: i) the boundary is an irreducible Q-Cartier divisor, ii) the general point of the boundary is smooth, iii) the codimension of A g . We point out that in this case for r ≤ 4 the general fibre of q : A (r) g → A g−r has dimension gr − r(r + 1)/2, thus dim(A (r)) g ) = g(g + 1)/2 − r. We define for our chosen compactificationÃ g the boundary as ∂Ã g := A For the Voronoi compactification the fibres of the map q : correspond to socalled stable quasi-abelian varieties which are compactifications of semi-abelian varieties, which can be explicitely described (see [29,28,5] and § §15, 17 for torus ranks 1 and 2). Thus one can define the vertical singular locus and the Andreotti-Mayer loci on the partial compactification A ′ g (see Remark 18.1 below). For higher torus rank the situation is more complicated. For instance the fibres of q : A (≥r) g → A * g−r might be non-reduced. However we will not need r > 4 here. An alternative approach might be to use the idea of Alexeev and Nakamura (cf. [1], [28], [5]) who have constructed canonical limits for 1-dimensional families of abelian varieties equipped with principal theta divisors. The stable quasi-abelian varieties that occur in A ′ g for torus rank 1 and 2 are exactly those described in Section 15 and 17. For torus rank 3 these are described by the tesselations of R 3 occuring on p. 188 of [28], cf. also Remark 17.3. The open stratum of A (3) g over A g−3 corresponds to the standard compactifications (see §16), obtained by glueing six P 3 -bundles over a (g − 3)-dimensional abelian variety B generalizing the construction for torus rank 2 where two P 2 -bundles were glued. These closed strata correspond to degenerations of the matrix T of the glueing data on which the standard compactifications depend (see §16). For instance the codimension 3 stratum corresponds to the fact that in T two of the three elements above the main diagonal tend to zero (or to ∞). Remark 18.1. As we remarked before, for stable quasi-abelian varieties corresponding to points (X, Ξ) of A ′ g one can define the vertical singularities Sing vert (Ξ) using Ω 1 (log D) as in the previous sections. One checks that for these compactifications the analogue of Lemma 16.1 still holds.
We will need the following result from [14].
Theorem 18.2. Let Z be an irreducible, closed subvariety ofÃ g . Then Z ∩∂Ã g is not empty as soon as codimÃ g (Z) < g.

The Andreotti-Mayer loci and the boundary
We are working with a fixed compactificationÃ g =Ã Vor g and, as indicated in §18 above, we can define the Andreotti-Mayer loci over the part A ′ g = A (≤4) g of A g . We have N g,k as a subscheme of A g and we defineÑ g,k as the schematic closure. The support ofÑ g,k contains the set of points corresponding to pairs (X, Ξ) such that Sing vert (X) has a component of dimension at least k (see [27]).
It is interesting to look at the case k = 0, which has been worked out by Mumford [27] and Debarre [11]. In this caseÑ g,0 is a divisor and by Theorem 18.2, every irreducible component N of this divisor intersects ∂Ã g . Let M be an irreducible component of N ∩ ∂Ã g , which has dimension g+1 2 − 2.
First of all, notice that M cannot be equal to A (≥2) g . This follows by the results in §17 and by Propositions 11.6 and 12.1. More generally, in the same way, one proves that M cannot contain A If M dominates A g−1 via q, the fibre of q |M over a general point (B, Ξ) ∈ A g−1 is N 0 (B, Ξ). Recall now that Debarre proves in [11] that N 0,g has two irreducible components, one of which is the so-called theta-null component θ 0,g : the general abelian variety (X, Θ X ) in θ 0,g , with Θ X symmetric, is such that Θ X has a unique double point which is a 2-torsion point of X lying on Θ X . Let M 0,g be the other component. The general abelian variety (X, Θ X ) in M 0,g , with Θ X symmetric, is such that Θ X has exactly two double points x and −x. It is useful to recall that, by Corollary 8.12, at a general point of either one of these component of N g,0 , the tangent cone to the theta divisor at the singular points is a smooth quadric. The component θ 0,g intersects the boundary in two components, θ ′ 0,g , θ ′′ 0,g , one dominating θ 0,g−1 , the other dominating A g−1 with fibre over the general point (B, Ξ) ∈ A g−1 given by the component 2Ξ of N 0 (B, Ξ) (see §11). Also M 0,g intersects the boundary in two irreducible divisors M ′ 0,g , M ′′ 0,g . The former is irreducible and dominates M 0,g−1 , the latter dominates A g−1 with fibre over the general point (B, Ξ) ∈ A g−1 given by the components of N 0 (B, Ξ) different from 2Ξ. The main ingredient for Debarre's proof of the irreducibility of M 0,g is a monodromy argument which implies that, if (B, Ξ) is a general principally polarized abelian variety of dimension g, then N 0 (B, Ξ) consists of only two irreducible components.
Remark 19.1. Let (B, Θ B ) be a general element in θ 0,g and let (X, Ξ) be a semi-abelian variety of torus rank one with abelian part B. Then there are no points in Sing vert (Ξ) with multiplicity larger than 2. This follows from the fact that Θ B has a unique singular point and by Remark 15.1.
We finish this section with the following result which will be useful later on. It uses the notion of asymptotic cone given in §6. Proof. Degenerate to the jacobian locus and apply the results from [18,19].
20. The Conjecture for N 1,g In this section we prove Conjecture 9.1 for k = 1. We consider an irreducible component N ofÑ g,1 which is of codimension 3. The first observation is that the assumption about the codimension of N implies that the generic principally polarized abelian variety is simple since by Remark 7.4 every irreducible component of A (ns) g has codimension ≥ g − 1 > 3 if we assume g ≥ 5.
Proposition 20.1. Let g ≥ 6 and let N be an irreducible component ofÑ g,1 which is of codimension 3 inÃ g . Then N intersects the stratum A Proof. We begin by remarking that N cannot be complete in A g in view of Theorem 18.2. Therefore N intersects ∂Ã g . Here we shall use the perfect cone compactification, see §18. Since ∂Ã g is a divisor inÃ g it intersects N in codimension one. Let M be an irreducible component of N ∩ ∂Ã g . It is our intention to prove that M has a non-empty intersection with A g . Then either M maps dominantly toÃ g−3 via the map q : A (3) g Ã g−3 and M intersects the general fibre F of q in a divisor, or M maps to a divisor in A g−3 under q with full fibres F contained in M . The former case is impossible by Proposition 12.1. In the latter case for a general (B, Ξ) in q(M ) all the quasi-abelian varieties (X, Ξ) in the fibre F over (B, Ξ) must have a 1-dimensional vertical singular locus of Ξ. Note that (X, Ξ) corresponds to a standard compactification as considered in §16. By the discussion given in §16 and by Proposition 11.6, we see that q(M ) has to be contained in N g−3,1 , against Theorem 8.6. We thus conclude that M ∩ A g has codimension 2 in A g → A g−2 over (B, Ξ). By taking into account the analysis §17 and Proposition 11.6, this implies q(M ) contained in N g−2,1 , giving again a contradiction. This proves that M ∩ A From now on we use again the Voronoi compactification. Let g ≥ 4 and let N be an irreducible component of N g,1 of codimension 3 in A g . As in the proof above we denote by M an irreducible component of the intersection of the closure of N inÃ g with the boundary ∂Ã g . According to Lemma 20.1 the morphism q :Ã g → A * g induces a rational map α : M Ã g−1 , whose image is not contained in ∂Ã g−1 .
Lemma 20.2. In the above setting, the Zariski closure of q(M ) inÃ g−1 is: (i) either an irreducible component N 1 ofÑ g−1,1 of codimension 3 inÃ g−1 ; (ii) or an irreducible component N 0 ofÑ g−1,0 and in this case: the singular locus D of X in one or two points, whose associated quadric has corank 1.  (N 1 (B, Ξ)) ≥ 2 and since N g−1,0 is a divisor in A g−1 we see that (iia) follows. The last statement (iib) follows from Remark 12.3, the analysis in Section 17, the description of N g,0 by Mumford and Debarre (see [27], [11], and §19) and Corollary 8.12.
We are now ready for the proof of the conjecture for N g,1 . (ii) g = 5 and N = J 5 is the jacobian locus.
Proof. By Theorem 8.6, the codimension of N is at least 3. Suppose that N has codimension 3. It is well-known that the assertion holds true for g = 4 and 5 (see [4], [10], [8]). We may thus assume g ≥ 6 and proceed by induction. Let ζ = (X, Θ X ) be a general point of N , so that X is simple (see Remark 7.4). Let S be a 1-dimensional component of Sing(Θ X ). We can assume that the class of S in X is a multiple mγ X of the minimal class γ X = Θ g−1 X /(g − 1)! ∈ H 2 (X, Z). If not so, then End(X) = Z and this implies that codim Ag (N ) ≥ g − 1 (see Remark 7.4). By Theorem 8.6, the general point in S is a double point for Θ X . We let R be the curve in S g whose general point is ξ = (X, Θ X , x), with x ∈ S a general point. Note that R is birationally equivalent to S. Let Q be the linear system of all quadrics in P(T X,0 ). One has the map φ : ξ ∈ R Q ξ ∈ Q.
As in the proof of Theorem 8.6, the map φ is not constant. Let Q R be the span of the image of φ. As in the proof of Theorem 8.6, one has dim(Q R ) ≥ 2. By Proposition 8.3, Q R is contained in the linear system N g,1 (ζ) (see §7 for the definition), which has dimension at most 2 since codim Ag (N ) = 3. Thus Q R = N g,1 (ζ) has dimension 2. By Lemma 20.1, the closure of N in A Proof of the claim. By induction, one reduces to the case g = 6 and α(M ) = J 5 . Let (X, Ξ) ∈ M be a general point. Then (X, Ξ) is a general rank one extension of the Jacobian (J(C), Θ C ) of a general curve C of genus 5. Note that if x ∈ J(C) corresponds to the extension, then Θ C and x + Θ C are not tangentially degenerate (see [17], Thm. 10.8, p. 273). Then the analysis of §15 implies that the vertical singular locus S 0 of Ξ sits on the singular locus D ∼ = J(C) of X and it is isomorphic to S C = Sing(Θ C ) with cohomology class Θ 4 /12 (see [3]). Thus Ξ · S 0 = Θ C · S C = 10. Hence, if ζ = (X, Θ X ) is a general point of N , then Sing(Θ X ) is a curve S such that Θ X · S = 10. On the other hand S is homologous to mγ X and one has 10 = mΘ X · γ X = 6m, a contradiction.
Claim 20.4 shows that only possibility (ii) in Lemma 20.2 can occur. In particular, by (iib) of Lemma 20.2, for ξ = (X, Θ X , x) general in R, the quadric Q ξ has corank 1. Let v ξ ∈ P g−1 be the vertex of Q ξ . Remember that R is birational to S. Hence, by Proposition 4.4, the map can be regarded as the Gauss map γ S of S.
Claim 20.5. If the general quadric in the linear system Q R is singular, then for ξ = (X, Θ X , x) general in R, the vertex v ξ of Q ξ is contained in the asymptotic cone T C Proof of the claim. Suppose this is not the case. Again we consider an irreducible component M ofN ∩ ∂Ã ′ g and let (X, Ξ) be a general point of M . By Claim 20.4 and Lemma 20.2, (X, Ξ) is a rank 1 extension of (B, Ξ) corresponding to a general point in a component of N g−1,0 , with extension datum b varying in a codimension 2 component of N 1 (B, Ξ). We let S 0 be the vertical singular locus of Ξ. By the analysis of §15, this corresponds to a contact curve C := C b of Ξ with Ξ b , which contains the singular points of both Ξ and Ξ b (see Remark 12.3). This means that we have a point on S 0 , corresponding to a singular point x of Ξ, where the tangent cone is the cone over Q x with vertex the point of P g−1 corresponding to b (see S 15). Now we note that C is smooth at x. Indeed locally around x, the divisor Ξ looks like a quadric cone of corank 1 in P g−2 and Ξ b looks like a hyperplane, which touches it along a curve. This implies that C locally at x looks like a line along which a hyperplane touches a quadric cone of corank 1 (see Remark 12.4). Hence, the Gauss image x b := γ C b (x) lies in Q x and actually, by Claim 20.5, the point x b lies in the asymptotic cone T C (4) x . The Gauss map γ Ξ : Ξ P g−2 of Ξ has an indeterminacy point at x and −x, which can be resolved by blowing up x and −x, since we may assume Ξ to be symmetric. Let p :Ξ → Ξ be the blow-up and letγ Ξ be the morphism which coincides with γ Ξ • p :Ξ P g−2 on an open subset. The exceptional divisor at x and −x is isomorphic to Q x andγ Ξ (x b ) is the tangent hyperplane to Q x at x b . The tangency property of Ξ and Ξ b along C implies that the tangent hyperplane to Q x at x b coincides with γ Ξ b (x). Now we let b vary in a component Z of N 1 (B, Ξ) of dimension g − 3, so that we have a rational map Note that Q x also has dimension g − 3 and we claim that f has finite fibres, hence it is dominant. If not, we would have an irreducible curve Γ in Z such that for all b ∈ Γ, x b stays fixed. But then for the general b ∈ Γ, the divisor Ξ b has a fixed tangent hyperplane at x, a contradiction by Lemma 11.1. On the other hand, by Claim 20.5 and by part (i) of Proposition 19.2, one has that f cannot be dominant, a contradiction.
By Lemma 20.6, the curve φ(R) := Σ is an irreducible component of the discriminant ∆ ⊂ Q R of singular quadrics in Q R ∼ = P 2 . Note that ∆ has degree g in P 2 . The map φ has degree at least 2 since we may assume Θ X to be symmetric, hence it factors through the multiplication by −1 X on X.
Proof. Let d ≥ 2 be the degree of the map. Then for ξ = (X, Θ X , x) general in R, we have distinct points ξ i = (X, Θ X , x i ) ∈ R, with ξ = ξ 1 , such that all the quadrics Q ξi , i = 1, . . . , d, coincide with the quadric Q = Q ξ . If for i = 1, . . . , d we let η i be the tangent vector to S at x i , we have that η 1 = · · · = η d and ∂ ηi Q ξi , for i = 1, . . . , d, coincide with the quadric Q 1 = ∂ η1 Q, which is linearly independent from Q. The analysis we made in §3 shows that S g is smooth at each of the points ξ i , i = 1, . . . , d, and the tangent space there is determined by Q and Q η1 . This implies that a general deformation of ξ = (X, Θ X , x) inside S g carries with it a deformation of each of the points ξ i (i = 1, . . . , d) in S g , because the involved quadrics are the same at these points. This yields that a general element of an irreducible component of N g,0 containing N has at least d singular points. By Debarre's result in [11] (see §19), one has d = 2, proving our claim.
Claim 20.8. The map φ is a morphism.
Proof. To prove the claim, it suffices to show that N is not contained in N g,0,r , with r ≥ 3. In order to prove this, one verifies that, for a general point ( Note now that the morphism φ is defined on R ∼ = S by sections of O S (Θ X ), since the points of S verify the equations (2) and, if ξ = (X, Θ X , x) ∈ R, the entries of the matrix of Q ξ are ∂ i ∂ j θ(τ, z), where z corresponds to x. We deduce from deg(∆) = g and from Claim 20.7, that (31) S · Θ X ≤ 2g.
As we assumed at the beginning of the proof, the class of S in X is a multiple mγ X of the minimal class. In view of (31), we find m ≤ 2. The Matsusaka-Ran criterion [31] and a result of Welters [40] imply that (X, Θ X ) is either a Jacobian or a Prym variety or depends on less than 3g parameters. Since g ≥ 6 this is not possible in view of the dimensions. This ends the proof.
Remark 20.9. A. Verra communicated to us an interesting example of an irreducible component M of codimension 6 of N 6,1 contained in the Prym locus. We briefly sketch, without entering in any detail, its construction and properties. Let C be the normalization of a general curve of type (4, 4) on P 1 × P 1 with two nodes on a line of type (0, 1), so that C has genus g = 7.
Let d, t be the linear series formed by pull-back divisors on C of the rulings of type (1, 0), (0, 1) respectively. Consider a non trivial, unramified double cover f :C → C of C and let (P, Ξ) be the corresponding Prym variety. Then Ξ has a 1-dimensional unstable singular locus R (see [23]), homologically equivalent to twice the minimal class, described by all classes in Pic 12 (C) of divisors of the form f * (d) + M , with f * (M ) ∈ t. One proves that the map φ described in the proof of Theorem 20.3 sends R to a plane sextic of genus 7 which is tetragonally associated to C (see [23] for the tetragonal construction). The divisor Ξ has 24 further isolated singular points, which are pairwise exchanged by the multiplication by −1. One shows that the corresponding 12 tangent cones span the same linear system Q of dimension 2 spanned by φ(R). The linear system Q is the tangent space to the Prym locus P 6 at (P, Ξ), which is therefore a smooth point for P 6 . By contrast, M is a non-reduced component of N 6,1 of codimension 6 such that the projectivized normal space Q at a general point has dimension 2 rather than 5. This shows that the hypotheses of Theorem 20.3 cannot be relaxed by assuming only that the projectivized normal space to M at a general point has dimension 2.

Appendix: A Result on Pencils of Quadrics
One of the ingredients of the proof Theorem 8.6 is a classical result of Corrado Segre from [33] on pencils of quadrics. First we recall the following: Proposition 21.1. Let L be a pencil of quadrics in P n with n ≥ 1 whose general member is smooth. Then: (i) the number of singular quadrics Q ∈ L is n+1, where each such quadric Q has to be counted with a suitable multiplicity µ(Q) ≥ n + 1 − rk(Q); (ii) for a singular quadric Q ∈ L one has µ(Q) ≥ 2 if and only if either rk(Q) < n or the singular point of Q is also a base point of L; (iii) for a singular quadric Q ∈ L with rank n one has µ(Q) = 2 if and only if any other quadric Q ′ ∈ L is smooth at p and the tangent hyperplane to Q ′ at p is not tangent to Q along a line.
Proof. Consider the linear system Q n of dimension n(n + 3)/2 all quadrics in P n . Inside Q n we have the discriminant locus ∆ n of singular quadrics, which is a hypersurface of degree n + 1, defined by setting the determinant of a general quadric equal to zero. The differentiation rule for determinants implies that the locus ∆ n,r of quadrics of rank r < n + 1 has multiplicity n + 1 − r for ∆ n . By intersecting ∆ n with the line corresponding to L we have (i). As for assertion (ii), we may assume rk(Q) = n, so that Q has a unique double point p, which we may suppose to be the point (1, 0, . . . , 0). Thus the matrix of Q is of the form 0 0 n 0 t n A where 0 n ∈ C n is the zero vector and A is a symmetric matrix of order n and maximal rank. Let Q ′ be another quadric in L, with matrix with β ∈ C, b = (b 1 , . . . , b n ) ∈ C n and B is a symmetrix matrix of order n. By intersecting L with ∆ n , we find the equation The constant term in the left-hand-side is 0. The coefficient of the linear term is which proves (ii). Let us prove (iii). Suppose rk(Q) = n, so that Q has a unique double point p, which is a base point of L. Again we may suppose p is the point (1 : 0 : . . . : 0) and we can keep the above notation and continue the above analysis. The left-hand-side in (32) is hence the coefficient of the third order term is One has µ(Q) = 2 if and only if this determinant is not zero, hence b = 0, which is equivalent to saying that all quadrics in the pencil different from Q are smooth at p. Note that there is a vector c = (c 1 , . . . , c n ) ∈ C n such that b = c · A. Now the determinant in (33) vanishes if and only if c · b t = 0, i.e. c · A · c t = 0. This means that the line L joining p with (0 : c 1 : . . . : c n ) sits on Q and that the tangent hyperplane to Q ′ at p, which has equation b 1 x 1 + · + b n x n = 0, is tangent to Q along r.
Next we prove Segre's theorem.
Theorem 21.2. Let L be a linear pencil of singular quadrics in P n with n ≥ 2 whose general member Q has rank n + 1 − r, i.e. Vert(Q) ∼ = P r−1 . We assume that Vert(Q) is not constant when Q varies in L with rank n + 1 − r. Then: (i) the Zariski closure is a variety of dimension r spanning a linear subspace Π of dimension m in P n with r ≤ m ≤ (n + r − 1)/2; (ii) V L is a variety of minimal degree m − r + 1 in Π; (iii) if dim Q∈L, rk(Q)=n+1−r Vert(Q) = s then r ≤ (n + 2s + 3)/3; (iv) the number of quadrics Q ∈ L of rank rk(Q) < n + 1 − r is n + r − 2m − 1 ≤ n − r − 1, where each such quadric Q has to be counted with a suitable multiplicity ν(Q) ≥ n + 1 − r − rk(Q).
Q ∈L with Λ is singular at p ∈ Λ, but Q is not singular at p. Then Λ is tangent to Q at p and therefore also intersects the vertex of Q.
In conclusion we have only two possibilities for getting singular quadrics inL: (a) there is quadric of rank n + 1 − r in L whose vertex intersects Λ; (b) there is quadric of rank n + 1 − h < n + 1 − r in L giving rise to a quadric of rank n − h inL.
Case (a) occurs as many times as the degree of V L , that is, m − r + 1 times. According to part (ii) of Proposition 21.1, each quadric Q in case (a) contributes with multiplicity at least 2 in the counting of singular quadrics inL. We claim that, because of the generality of Λ, this multiplicity is exactly 2. To prove this, by part (iii) of Proposition 21.1, we will prove that for each quadric Q in case (a), with vertex p ∈ Λ and for any other quadric Q ′ ∈ L, the tangent hyperplane π to Q ′ at p is not tangent to Q along a line contained in Λ. To see this we can, by first cutting with a general subspace of dimension n − r + 1 through Λ, reduce ourselves to the case r = 1, in which V L is a rational normal curve. Choose then a general point q ∈ V L and let Q ′ = Q q be the unique quadric in L with a double point at q. The hyperplane π is tangent to Q q along the line p, q . This implies that π is tangent to Q only along the tangent line L p to V L at p (see the proof of part ii)). By the generality assumption, Λ is not tangent to V L at p. Thus the assertion follows.
As for quadrics in case (b), again by part (i) of Proposition 21.1, each such quadric contributes to the same count with multiplicity h − r. Since, by part (i) of Proposition 21.1, the number of singular quadrics inL, counted with appropriate multiplicity, is n − r + 1, the assertion follows.
One has the following consequence: Corollary 21.3. Let L be a linear pencil of quadrics in P n with n ≥ 2. Then the general member Q ∈ L has rank n + 1 − r (i.e., Vert(Q) ∼ = P r−1 ) if and only if the base locus of L contains a linear subspace Π of dimension m with r ≤ m ≤ (n + r − 1)/2, along which all the quadrics in L have a common tangent subspace of dimension n + r − m − 1. In this case Π is the span of the variety V L .
Proof. As usual it suffices to prove the assertion for r = 1. If the general quadric Q ∈ L has rank rk(Q) = n, the assertion follows from the proof of Theorem 21.2. The converse is trivial, since a smooth quadric in P n has a tangent subspace of dimension n − m − 1, and not larger, along a subspace of dimension m.
These results imply the existence of canonical forms for pencils of singular quadrics, originally due to Weierstrass [38] and Kronecker [21]. This is explained in some detail in [33], § §20-25, and we will not dwell on this here. It would be desirable to have an extension of the results in this Appendix to higher-dimensional linear families of quadrics.