# Sebastian Casalaina-Martin

## Prym varieties and the Schottky problem for cubic threefolds.

ABSTRACT: A theorem of Mumford's states that for a smooth cubic
threefold
X, the intermediate Jacobian JX is a principally polarized abelian
variety of dimension 5 whose theta divisor has a unique singular
point,
which has multiplicity three. This talk describes joint work with R.
Friedman, in which we prove a converse: if A is a principally
polarized
abelian variety of dimension 5 whose theta divisor has a unique
singular
point, which has multiplicity three, then A is the intermediate
Jacobian
of a smooth cubic threefold.
The method of proof is to view A as a generalized Prym variety and
to
use this description to analyze the singular points of the theta
divisor.
Along these lines, I will also discuss recent work which gives a sharp
upper bound on the multiplicity of a point on the theta divisor of an
irreducible principally polarized abelian variety of dimension at most
five.