# What is the closure of product loci in A_g?

Let $A_g$ denote the moduli space of principally polarized abelian varieties of dimension $g$. For any partition of $g$, one can consider the corresponding locus inside $A_g$ of products of lower-dimensional abelian varieties.

My question is: can one describe explicitly the closure of this locus in (some) compactification?

I guess part of this question is what compactification one should use. For instance, if one simply chooses a toroidal compactification $\widetilde{A}_g$, one can also extend the universal family to a family of semistable abelian varieties. Can one describe necessary and sufficient conditions for a semistable abelian variety at the boundary for it to be in the closure of such a product locus? Or perhaps if one restricts attention to rank-one degenerations, so it does not depend on a choice of toroidal compactification?

Alternatively, one could choose a point in the closure of $A_g$ inside Alexeev's space of stable semi-abelic pairs -- are there necessary and sufficient conditions in this situation?

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Of course a ppav in A(g) is a product iff its theta divisor is singular in codimension one. Maybe that has a generalization to semi abelian varieties. – roy smith Jan 10 '11 at 23:05
@roy smith: The most obvious generalization does not work. The semi-abelic varieties of Alexeev are typically singular in codimension $1$, hence every Cartier divisor (e.g., the theta divisor) is singular in codimension $1$. – jlk Jan 12 '11 at 20:17

This isn't really an answer, but perhaps a suggestion for where to get some ideas. Michael van Opstall studied moduli spaces of products of stable curves (of general type) in this and this papers. While his focus is on curves of genus $g>1$, some of the general deformation theoretic arguments might help you. Also, after all, a polarized abelian variety is of log general type, so possibly the arguments in this paper can be extended to pairs and then applied à la Alexeev.