Corrigendum To:

“Andreotti–Mayer Loci and the Schottky Problem”

cf. Documenta Math. 13 (2008) 398–440

Ciro Ciliberto, Gerard van der Geer

Received: January 29, 2014

Communicated by Thomas Peternell

Abstract. The main result of [2] (C. Ciliberto, G. van der Geer: Andreotti-Mayer Loci and the Schottky problem. Documenta Math. 13 (2008), 389–440) is based on a proposition whose proof is incorrect. We therefore give an alternative proof of the main result of [2].

2010 Mathematics Subject Classification: 14K10

Keywords and Phrases: Theta divisor, Schottky problem, Andreotti-Mayer loci

1. Introduction

In this note we make some corrections to [2]. It was pointed out to us by Grushevsky and Hulek that the proof of Proposition 12.1 of loc. cit. is not correct; see also [4]. Since this proposition is used in the proof of the main result of [2] (the proof of Conjecture 9.1 in the case N_{g,1}) we will give an alternative proof of this result that does not use Proposition 12.1. In this proof we make essential use of the analysis in [4]. We refer to [2] for the notation we use. We thank Grushevsky and Hulek for spotting the error in the proof and for subsequent correspondence.

2. The Conjecture for N_{g,1}

Recall from [2] that N_{g,k} ⊂ A_{g} is the Andreotti-Mayer locus consisting of isomorphism classes of principally polarised abelian varieties (B, Ξ) such that the singular locus of Ξ has dimension ≥ k. The main result [2, Thm 20.3] of [2] is:

Theorem 2.1. Let N be an irreducible component of N_{g,1} with g ≥ 4, of codimension 3 in A_{g}. If g = 4, assume also that the general point of N is not a product of an elliptic curve with a 3-dimensional abelian variety. Then either N = J_{5}, the jacobian locus in A_{5}, or N = H_{4}, the hyperelliptic locus in A_{4}.

The proof of Theorem 2.1 given in [2] is based on [2, Proposition 20.1] and [2, Lemma 20.2], in which we use [2, Proposition 12.1]. The proof of [2, Proposition 12.1] is wrong. The proposition is a generalization of [1, Prop. 3]. Since the proof of [2, Thm. 20.3] depends on it, this proof is therefore faulty. We give here a slightly different proof of [2, Proposition 20.1] and [2, Lemma 20.2], though the basic idea is the same as in [2], but we have to consider other types of degenerations of abelian varieties. The statement of the theorem is well-established for g = 4 and g = 5, see the references in [2]. We therefore assume in the following that g ≥ 6. We begin by recalling the notation. For a principally polarized abelian variety (B, Ξ) we define N_{k,1}(B, Ξ) to be the locus of u ∈ B such that singular locus of the scheme-theoretic intersection Ξ · Ξ_{u} has dimension at least 1. By Ã_{g} we mean the perfect cone compactification of A_{g}. This is a stratified space with strata A_{g}^{(r)}, with A_{g}^{(r)} the stratum of quasi–abelian varieties of torus rank r, with closure A_{g}^{(≥r)}. Note that ∂Ã_{g} := Ã_{g} − A_{g} = A_{g}^{(≥1)} is the ‘boundary’ of Ã_{g}. The stratum A_{g}^{(1)} is common to all toroidal compactifications of A_{g}, so also to the second Voronoi compactification. Finally, by Ñ_{g,k} we mean the closure of the Andreotti–Mayer locus N_{g,k} in Ã_{g}.

The statement of [2, Proposition 20.1], for which we will give a new proof, is the following:

Proposition 2.2. Let g ≥ 6 and let N be an irreducible component of Ñ_{g,1} of codimension 3 in Ã_{g}. Then N ∩ A_{g}^{(1)} ≠ ∅.

Proof. By [3], N is not complete in A_{g}. Therefore N ∩ ∂Ã_{g} ≠ ∅. We shall show that the closure of N ∩ A_{g} in Ã_{g} intersects Mumford’s partial toroidal compactification A_{g}^{(≤1)} inside Ã_{g}. The reason for using the perfect cone compactification is the fact that the locus A_{g}^{(≥r)} has codimension r in it. Since ∂Ã_{g} is a divisor in Ã_{g}, it intersects N in codimension one. Let M be any irreducible component of N ∩ ∂Ã_{g}. We make a case distinction. Step 1. Suppose first M ⊆ A_{g}^{(≥4)}. For dimension reasons we then have M = A_{g}^{(≥4)}. So the general point of M corresponds to a general standard compactification (\overline{X}, \overline{Ξ}) of torus rank 4 (see [2, §16]) with abelian part (B, Ξ) general in A_{g−4}. Recall that (\overline{X}, \overline{Ξ}) corresponds to the datum of a point b = (b_{1}, …, b_{4}) ∈ B^{4} and a 4×4 matrix T = (t_{ij}) with entries in ℂ^{∗} such that t_{ii} = 1 and t_{ij} = t_{ji}^{−1}. By letting some t_{ij} with i ≠ j tend to 0 (or equivalently to ∞) we obtain special points of M = A_{g}^{(4)}, see [2, §17] for a similar discussion in the case of torus rank 2. We now write down a local equation for the generalized theta divisor. We let z = (z_{1}, …, z_{g−4}) ∈ ℂ^{g−4} be coordinates on the universal cover of B, and ξ(z) = 0 the equation of Ξ. Given ζ = (ζ_{1}, …, ζ_{4}) ∈ ℂ^{4}, we set u_{i} = exp(2πζ_{i}) for 1 ⩽ i ⩽ 4, u = (u_{1}, …, u_{4}), and u_{I} = ∏_{i∈I} u_{i}. Finally, we set t_{I} = ∏_{i,j∈I,i≤j} t_{ij} and b_{I} = ∑_{i∈I} b_{i} and let ω_{I} ∈ ℂ^{g−r} represent b_{I} ∈ B. We use the convention t_{∅} = 1, b_{∅} = 0, ω_{∅} = 0. The generalized theta divisor \overline{Ξ} of \overline{X} as in [2, §16] is given by the equation

(1)  𝔵(z, u, b, T) = ∑_{I⊆{1,...,4}} u_{I} t_{I} ξ(z − ω_{I}) = 0 .

By the hypothesis, the system

(2)  \begin{matrix}𝔵 = & 0 \\ u_{h}∂_{u_{h}}𝔵 = & 0 & for 1 ⩽ h ⩽ 4 \\ ∂_{z_{i}} 𝔵 = & 0 & for 1 ⩽ i ⩽ g − 4 \end{matrix}

has an at least 1–dimensional set of solutions in z, u for all b and T and (B, Ξ) ∈ A_{g−4} general. As pointed out above, in (1) we may even take t_{ij} = 0 for all pairs (i, j) with i < j and b_{i} = b ∈ B, and accordingly ω_{i} = ω, for all i with 1 ≤ i ≤ 4. Then (2) reads

(3)  \begin{matrix}ξ(z) + (∑_{i=1}^{4} u_{i})ξ(z − ω) = & 0 \\ u_{h}ξ(z − ω) = & 0 & for 1 ⩽ h ⩽ 4 \\ ∂_{z_{i}}ξ(z) + (∑_{i=1}^{4} u_{i})∂_{z_{i}}ξ(z − ω) = & 0 & for 1 ⩽ i ⩽ g − 4 \end{matrix}

which still has an at least 1-dimensional set of solutions in z, u for all b ∈ B and general (B, Ξ) in A_{g−4}. If for all these solutions one has ∑_{i=1}^{4} u_{i} = 0, then (3) implies

ξ(z) = ∂_{z_{i}}ξ(z) = 0 for 1 ⩽ i ⩽ g − 4

implying (B, Ξ) ∈ N_{g−4,1}, a contradiction, because (B, Ξ) ∈ A_{g−4} is taken general. So we may assume ∑_{i=1}^{4} u_{i} ≠ 0; in particular, there is an h with 1 ≤ h ≤ 4 such that u_{h} ≠ 0. Then (3) reads

ξ(z) = 0, ξ(z − ω) = 0, ∂_{z_{i}}ξ(z) + (∑_{i=1}^{4} u_{i})∂_{z_{i}}ξ(z − ω) = 0 for 1 ⩽ i ⩽ g − 4 .

This implies that N_{1,1}(B, Ξ) = B, which is impossible by [2, Proposition 11.6] since (B, Ξ) ∈ A_{g−4} is general. Or, alternatively, [1, Proposition 3] implies (B, Ξ) ∈ N_{g−4,0}, again a contradiction.

Remark 2.3. In the argument here we used specialization. This may bring us into a deeper stratum. But the local equation for the generalized theta divisor in the deeper stratum is a limit of the local equation of the generalized theta divisor. Similarly the limits of the equations (2) that describe the singularities gives the singular locus in the limit. Since the flat limit of the solution space is contained in the set of solutions of the limit set of equations, we see that the singular locus has a dimension that is not less than the dimension of the singular locus in the generic fibre of our local family. This argument will be used repeatedly in what follows.

Step 2. Assume next M ∩ A_{g}^{(3)} ≠ ∅, so that M has codimension 1 in A_{g}^{(≥3)}. There are three possibilities. Either the general point of M corresponds to a standard compactification, or M coincides with one of the two strata of codimension 4 and torus rank 3 of the perfect cone decomposition. See the table on p. 1321 of [4]. Step 2a. We first deal with the first case. We have a map q : A_{g}^{(≥3)} −−→ Ã_{g−3}. derived from the map to the Satake compactification. We denote by F_{(B,Ξ)} the fibre of q over (B, Ξ) ∈ A_{g−3}. By Step 1, the rational map q is defined at the general point of M and q(M) ∩ A_{g−3} ≠ ∅. Then,

(1) either q_{|M} is dominant onto Ã_{g−3} and, for (B, Ξ) ∈ A_{g−3} general, M intersects F_{(B,Ξ)} in codimension 1;

(2) or q(M) has codimension 1 in Ã_{g−3} with full fibres contained in M.

As in Step 1, F_{(B,Ξ)} contains a dense open subset consisting of pairs (b, T), with b = (b_{1}, b_{2}, b_{3}) ∈ B^{3} and T a 3 × 3 matrix with entries in ℂ^{∗} such that t_{ii} = 1 and t_{ij} = t_{ji}^{−1}. We can obtain special points of F_{(B,Ξ)} as limits by letting some t_{ij}, with i ≠ j, tend to 0 (or, which is the same, to ∞), cf. Remark 2.3. Case (2) can be excluded by repeating the same argument as in Step 1 and taking into account that, by [2, Theorem 8.6], q(M) is not contained in N_{g−3,1}. (Note that the general point of q(M) corresponds to a simple abelian variety, so that [2, Proposition 11.6] can be applied; see [2, §7].) In case (1), we take (B, Ξ) ∈ A_{g−3} general. The fibre F_{(B,Ξ)} is fibered over B^{3} with fibres torus embeddings of 𝔾_{m}^{3}. The image L of M ∩ F_{(B,Ξ)} in B^{3} under the projection map to B^{3} is of codimension ≤ 1. First, suppose the codimension is 1. Since the fibre of M ∩F_{(B,Ξ)} → B^{3} contains the full fibre of p : F_{(B,Ξ)} → B^{3}, we can let the t_{ij} for i < j tend to zero and find the set of equations

(4)  \begin{matrix}ξ(z) + u_{1}ξ(z − ω_{1}) + u_{2}ξ(z − ω_{2}) + u_{3}ξ(z − ω_{3}) = 0 \\ u_{1}ξ(z − ω_{1}) = 0, u_{2}ξ(z − ω_{2}) = 0, u_{3}ξ(z − ω_{3}) = 0 \\ ∂_{z_{i}}ξ(z) + u_{1}∂_{z_{i}}ξ(z − ω_{1}) + u_{2}∂_{z_{i}}ξ(z − ω_{2}) + u_{3}∂_{z_{i}}ξ(z − ω_{3}) = 0 \end{matrix}

which for every triple (b_{1}, b_{2}, b_{3}) in the image has an at least 1-dimensional set of solutions in z and u_{i}. If for general (b_{1}, b_{2}, b_{3}) ∈ L we have u_{1} = u_{2} = u_{3} = 0, then B ∈ N_{g−3,1}, against our assumptions. If for general (b_{1}, b_{2}, b_{3}) in L exactly s (with 1 ≤ s ≤ 3) u_{i} are not identically zero then we see that the projection of L to B^{s} lies in N_{1,s}(B, Ξ). But this contradicts [2, Proposition 11.6], which says that the codimension in B^{s} is at least 2. (Alternatively, use [1, Proposition 3].) Second, we treat the case L = B^{3}. Then M ∩F_{(B,Ξ)} intersects the general fibre of p, which is a torus embedding of 𝔾_{m}^{3}, in a surface. Then we may let two t_{ij}, e.g., t_{13} and t_{23}, tend to 0, cf. Remark 2.3. Set t = t_{12}. Then we find the system (5)

ξ(z) + u_{1}ξ(z − ω_{1}) + u_{2}ξ(z − ω_{2}) + u_{3}ξ(z − ω_{3}) + tu_{1}u_{2}ξ(z − ω_{1} − ω_{2}) = 0

u_{i}(ξ(z − ω_{i}) + tu_{3−i}ξ(z − ω_{1} − ω_{2})) = 0, 1 ⩽ i ⩽ 2

u_{3}ξ(z − ω_{3}) = 0

∂_{z_{i}}ξ(z) + u_{1}∂_{z_{i}}ξ(z − ω_{1}) + u_{2}∂_{z_{i}}ξ(z − ω_{2}) + u_{3}∂_{z_{i}}ξ(z − ω_{3})

+ tu_{1}u_{2}∂_{z_{i}}ξ(z − ω_{1} − ω_{2}) = 0, 1 ⩽ i ⩽ g − 3

which for every triple (b_{1}, b_{2}, b_{3}) ∈ L = B^{3} and a suitable t has an at least 1dimensional set Z of solutions in z and u_{i}. If t is not constant, we may let t tend to 0 (or ∞ which amounts to the same), in which case we find a contradiction as before. Similarly if t = 0. So we may assume that t is a non–zero constant. If u_{1} = u_{2} = 0, then either (B, Ξ) ∈ N_{g−3,1} (if u_{3} = 0) or N_{1,1}(B, Ξ) = B (if u_{3} ≠ 0); the former case is not possible because q(M) = Ã_{g−3}, the latter is not possible by [2, Proposition 11.6]. So we may assume that one of the variables u_{1}, u_{2}, e.g., u_{1}, is not identically zero. Suppose u_{2} = 0. Since we are assuming (b_{1}, b_{2}, b_{3}) ∈ B^{3} general, we can specialize to b_{1} = b_{3}. Then (5) becomes

ξ(z) + (u_{1} + u_{3})ξ(z − ω_{1}) = 0
u_{1}ξ(z − ω_{1}) = u_{3}ξ(z − ω_{3}) = 0
∂_{z_{i}}ξ(z) + (u_{1} + u_{3})∂_{z_{i}}ξ(z − ω_{1}) = 0, 1 ⩽ i ⩽ g − 3

which implies that either (B, Ξ) ∈ N_{g−3,1} (if u_{1} + u_{3} = 0) or N_{1,1}(B, Ξ) = B (if u_{1} + u_{3} ≠ 0), both leading to contradictions. Thus we may assume that u_{1} and u_{2} are both not identically 0. Suppose that one of u_{1} and u_{2} is constant on the solution set Z of (5) for general (b_{1}, b_{2}) ∈ B^{2}. Let C be an irreducible curve in the projection of Z to B. By either the second or the third equation in (5) we have O_{C}(Ξ_{b_{i}} − Ξ_{b_{1}+b_{2}}) ≅ O_{C}. Since B is general, the map B → Pic(C) sending b to O_{C}(Ξ_{b} − Ξ_{b_{1}+b_{2}}) has finite kernel. Hence b_{i} has finite order, a contradiction, since b_{i} ∈ B is general. So u_{1} and u_{2} are non–constant. Then by taking the limit case where u_{2} = 0, the system (5) becomes like (4) with u_{2} = 0. If u_{1} and u_{3} are both non–zero, then N_{0,2}(B, Ξ) = B^{2}, contradicting [2, Proposition 11.6]. If one of u_{1} and u_{3} is zero, a similar argument works.

Step 2b. Here we assume that M coincides with the component of the perfect cone compactification that parametrizes compactifications of semi-abelian varieties that are a union of two ℙ^{1} × ℙ^{2}-bundles over an abelian variety B of dimension g − 3. This type has been described in [4, Section 7]. Here we find a theta divisor given on one component by an equation

\begin{matrix}ξ(z)+ & u_{1} ξ(z − ω_{1}) + u_{2} ξ(z − ω_{2}) + u ξ(z − ω_{3}) + \\ u_{1}u t_{13}ξ(z − ω_{1} − ω_{3}) + u_{2}ut_{23} ξ(z − ω_{2} − ω_{3}) = 0 . \end{matrix}

Again we have cases (1) and (2) as in Step 2a. Case (2) can be eliminated as above. In case (1) we can assume that t_{13} and t_{23} tend to 0. Then we find a system of equations

ξ(z) + u_{1}ξ(z − ω_{1}) + u_{2}ξ(z − ω_{2}) + uξ(z − ω_{3}) = 0
u_{1}ξ(z − ω_{1}) = 0, u_{2}ξ(z − ω_{2}) = 0, uξ(z − ω_{3}) = 0
∂_{z_{i}}ξ(z) + u_{1}∂_{z_{i}}ξ(z − ω_{1}) + u_{2}∂_{z_{i}}ξ(z − ω_{2}) + x∂_{z_{i}}ξ(z − ω_{3}) = 0

which has a 1-dimensional set of solutions in z and u_{1}, u_{2}, u (the affine coordinates on ℙ^{2} and ℙ^{1}) for all b_{i} ∈ B and (B, Ξ) ∈ A_{g−3} general. Note that the equations coincide with those of (4) and this case can be eliminated by a variant of the arguments used above. Step 2c. The final case left to be dealt with is the case where M coincides with the codimension 4 stratum of torus rank 3 which parametrizes compactifications whose toric part corresponds to a union of two ℙ^{3} and an intersection of two quadrics in ℙ^{5}, see [4, Section 7]. The case of a 1-dimensional singular locus on one of the two ℙ^{3} follows along the same lines as the cases above. For the intersection of quadrics, the theta divisor is described by an equation of the form (see [4])

(6)  \begin{matrix}ξ(z − ω_{1}) + u_{1}ξ(z − ω_{2} − ω_{3}) + tu_{2}ξ(z − ω_{2})+ \\ tu_{3}ξ(z − ω_{1} − ω_{3}) + su_{4}ξ(z − ω_{3}) + su_{5}ξ(z − ω_{1} − ω_{2}) = 0 \end{matrix}

where the ω_{1}, ω_{2}, ω_{3} can vary freely in ℂ^{g−2}. Again we can let s and t tend to 0 and then the arguments of Step 1 can be applied. Step 3. Next we may assume M ∩ A_{g}^{(2)} ≠ ∅, and then M has codimension 2 in A_{g}^{(≥2)}. Consider the dominant map q : A_{g}^{(≥2)} −−→ Ã_{g−2}, which, by Steps 1 and 2, is defined at the general point of M and we have q(M) ∩ A_{g−2} ≠ ∅. The fibre F_{(B,Ξ)} of q is now a compactified 𝔾_{m}-bundle over B^{2} with t := t_{12} the coordinate on 𝔾_{m}. There are three possibilities:

(1) either q_{|M} is dominant onto Ã_{g−2} and, for (B, Ξ) ∈ A_{g−2} general, M intersects F_{(B,Ξ)} in codimension 2;

(2) or q(M) has codimension 1 in Ã_{g−2} and, for (B, Ξ) ∈ q(M) general, M intersects F_{(B,Ξ)} in codimension 1;

(3) or q(M) has codimension 2 in Ã_{g−2} with full fibres contained in M. Case (3) can be excluded with the same argument in Step 1. Case (2) can be excluded with an argument similar to the one we gave in Step 2. Indeed, we take (B, Ξ) ∈ q(M) general. The image L of the projection of an irreducible component of M ∩ F_{(B,Ξ)} to B^{2} has either codimension 1 in B^{2} or L = B^{2}. The relevant system is now (7)

ξ(z) + u_{1} ξ(z − ω_{1}) + u_{2} ξ(z − ω_{2}) + tu_{1}u_{2} ξ(z − ω_{1} − ω_{2}) = 0

u_{1} ξ(z − ω_{1}) + u_{1}u_{2}t ξ(z − ω_{1} − ω_{2}) = 0,

u_{2} ξ(z − ω_{2}) + u_{1}u_{2}t ξ(z − ω_{1} − ω_{2}) = 0,

∂_{z_{i}}ξ(z) + u_{1} ∂_{z_{i}}ξ(z − ω_{1}) + u_{2} ∂_{z_{i}}ξ(z − ω_{2}) + u_{1}u_{2}t ∂_{z_{i}}ξ(z − ω_{1} − ω_{2}) = 0 .

with an at least 1–dimensional set of solutions Z.

If codim_{B^{2}}(L) = 1, then the full fibre of F_{(B,Ξ)} → B^{2} is contained in M ∩F_{(B,Ξ)}. Therefore we can let t = 0 in (7) and we can argue as in Step 2 (Case (1), codim_{B^{3}}(L) = 1). If L = B^{2}, we may assume, as above, that both u_{1}, u_{2} do not vanish identically on Z. By taking u_{i} = 0 in (7), we see that N_{0,1}(B, Ξ) = B, contradicting [2, Proposition 11.6]. Also case (1) can be treated in a similar way. We take (B, Ξ) ∈ A_{g−2} general. The image L of the projection of an irreducible component of M ∩ F_{(B,Ξ)} onto B^{2} has either codimension 1 or 2 in B^{2}. We let L_{1} (resp. L_{2}) be the projection of L on the first (resp. second) factor of B^{2}. If codim_{B^{2}}(L) = 2, then the full fibre of F_{(B,Ξ)} → B^{2} is contained in M ∩F_{(B,Ξ)}. Therefore we can let t = 0 in (7). As usual we may get rid of the case in which u_{1} = u_{2} = 0 on the set Z of solutions of (7). Assume next that only one among u_{1}, u_{2}, e.g. u_{1} vanishes identically. Then L_{2} ⊆ N_{1,1}(B, Ξ) and [2, Proposition 11.6] tells us that L_{2} = N_{1,1}(B, Ξ) has codimension 2 in B. But then N_{1,1}(B, Ξ) is positive-dimensional and Proposition [1, Proposition 3] yields (B, Ξ) ∈ N_{g−2,0}, again a contradiction. So we may assume that u_{1}, u_{2} both do not vanish identically on Z. By taking u_{i} = 0, we see that L_{i} ⊆ N_{0,1}(B, Ξ), for i ∈ {1, 2}. By [2, Proposition 11.6], we have L_{i} = N_{0,1}(B, Ξ), for 1 ⩽ i ⩽ 2, and L = L_{1} × L_{2}, hence L intersects the diagonal of B^{2}. Therefore we can take ω_{1} = ω_{2} = ω (corresponding to b ∈ N_{0,1}(B, Ξ) general) and t = 0 in (7), so that this system becomes

(8)  \begin{matrix}ξ(z) + (u_{1} + u_{2}) ξ(z − ω) = 0 \\ u_{1} ξ(z − ω) = 0, u_{2} ξ(z − ω) = 0 \\ ∂_{z_{i}}ξ(z) + (u_{1} + u_{2}) ∂_{z_{i}}ξ(z − ω) = 0, 1 ⩽ i ⩽ g − 2. \end{matrix}

If u_{1} + u_{2} = 0 identically on Z, we have (B, Ξ) ∈ N_{g,0}, a contradiction. If u_{1} + u_{2} is not identically 0, then (8) shows that N_{1,1}(B, Ξ) = N_{0,1}(B, Ξ) which is a divisor on B, contradicting [2, Proposition 11.6]. Finally, assume codim_{B^{2}}(L) = 1. As before we get rid of the case in which one of u_{1} and u_{2} vanishes identically on the solution set Z for some triple (b_{1}, b_{2}, t). If for some pair (b_{1}, b_{2}) ∈ L we can take the limiting case t = 0, we are also done. So we may assume that u_{1}, u_{2}, t are non–zero and t is constant. If u_{1} and u_{2} are constant on Z we find a contradiction as in Step 2a. So we may assume that u_{1} and u_{2} are both non–constant and again we can finish as in Step 2a. In conclusion M ⊈ A_{g}^{(≥2)}, hence M ∩ A_{g}^{(1)} ≠ ∅, proving the assertion.

□

Keeping the above notation, by Proposition 2.2 the map q : A_{g}^{(1)} −−→ Ã_{g−1} is defined at the general point of M. We finish by giving a new proof of [2, Lemma 20.2].

Lemma 2.4. In the above setting, the Zariski closure of q(M) in Ã_{g−1} is:

(i) either an irreducible component N_{1} of Ñ_{g−1,1} of codimension 3 in Ã_{g−1}; or

(ii) an irreducible component N_{0} of Ñ_{g−1,0} and if (B, Ξ) ∈ N_{0} is general, then M ∩ F_{(B,Ξ)} is an irreducible component of N_{1,1}(B, Ξ) of codimension 2 in B. Moreover, if ξ = (\overline{X}, \overline{Ξ}) ∈ M is a general point, then Sing_{vert}(\overline{Ξ}) meets the singular locus D of \overline{X} in one or two points, whose associated quadric has corank 1.

Proof. If q(M) ⊆ N_{g−1,1}, then [2, Theorem 8.6] implies 3 ≤ codim_{A_{g−1}}(q(M)) ≤ codim_{∂Ã_{g}}(M) = 3 and we are in case (i). If q(M) ⊈ N_{g−1,1}, the argument in the proof of Proposition 2.2 shows that q(M) ⊆ N_{g−1,0} and M ∩ F_{(B,Ξ)} ⊆ N_{1,1}(B, Ξ). By [2, Proposition 11.6] we have codim_{B}(N_{1,1}(B, Ξ)) ≥ 2 and since N_{g−1,0} is a divisor in A_{g−1} we are in case (ii). The final assertion follows just as in the proof of [2, Lemma 20.2, (ii)] that does not depend on [2, Proposition 12.1], and goes through without change. □

3. Further Corrections

In the proof of [2, Corollary 12.2] we erroneously refer to (a non–existing) Corollary 11.1. The reference should instead have been to Proposition 12.1, whose proof is incorrect. Hence the proof of Corollary 12.2 is faulty. However, this Corollary 12.2 has never been used in [2]. Proposition 12.1 of [2] is used in the proof of [2, Lemma 16.1], which is therefore incorrect. Lemma 16.1 is used to justify the final assertion of [2, Remark 18.1], which is however not used later. Remark [2, 12.3] is also based on the incorrect analysis of the proof of [2, Prop. 21.1]. We point out one more (inconsequential) error in [2, §17, p. 485, l. 22–23]. It is stated there that, for a general quasi–abelian variety (\overline{X}, \overline{Ξ}) of torus rank 2 with abelian part (B, Ξ), the point x ∈ \overline{X} − Sing(\overline{X}), corresponding to coordinates (u, z), is a vertical singularity of \overline{Ξ} if and only if z is singular for the divisor H of B defined by

ξ(τ, z − ω_{1})ξ(τ, z − ω_{2}) = t ξ(τ, z)ξ(τ, z − ω_{1} − ω_{2}).

Four lines below, we state that the existence of a vertical singularity of \overline{Ξ} implies that Ξ, Ξ_{b_{1}}, Ξ_{b_{2}} and Ξ_{b_{1}+b_{2}} are tangentially degenerate at some point of B. Both assertions are incorrect. If x as above is a vertical singularity, then z is singular for H, but in general the converse does not hold. Moreover, if the theta divisors Ξ, Ξ_{b_{1}}, Ξ_{b_{2}} and Ξ_{b_{1}+b_{2}} are tangentially degenerate at some point of B, this may give rise to a vertical singularity, but does not do so necessarily.

References

[1] A. Beauville, O. Debarre: Sur le problème de Schottky pour les variétés de Prym. Ann. Scuola Norm. Sup. Pisa Cl. Sci. (4) 14 (1987), p. 613–623 (1988).

Andreotti–Mayer Loci and the Schottky Problem 1001

[2] C. Ciliberto, G. van der Geer: Andreotti-Mayer Loci and the Schottky problem. Documenta Math. 13 (2008), p. 389–440.

[3] G. van der Geer: Cycles on the moduli space of abelian varieties. In: Moduli of Curves and Abelian Varieties. C. Faber, E. Looijenga, Ed., Aspects of Math. 33 (1999) Vieweg.

[4] S. Grushevsky, K. Hulek: Principally polarized semi-abelic varieties of small torus rank and the Andreotti-Mayer loci. Pure Appl. Math. Q. 7 (2011), Special Issue: In memory of Eckart Viehweg, 1309-1360.

Ciro Ciliberto Dipartimento di Matematica II Università di Roma Italy cilibert@mat.uniroma2.it Gerard van der Geer Korteweg-de Vries Instituut Universiteit van Amsterdam Postbus 94248 1090 GE Amsterdam The Netherlands geer@science.uva.nl 1002

Documenta Mathematica 19 (2014)