Abstract

Due to a famous result of Clemens and Griffiths a smooth cubic threefold $X$ is a unirational, but not rational variety. The key tool in their proof is the intermediate Jacobian $J(X)$, which is a principally polarized abelian 5-fold. Associating to a cubic threefold $X$ its intermediate Jacobian $J(X)$ defines a morphism $p: M_4 \to A_5$ from the moduli space $M_4$ of cubic threefolds to the moduli space $A_5$ of principally polarized abelian $5$-folds which, by the Torelli theorem, is injective.

We exhibit a suitable compactification $\overline{M}_4$ of $M_4$ such that the Torelli map extends to a morphism $\overline{p}: \overline{M}_4 \to \overline{A}_5$ to the second Voronoi compactification of $A_5$. In a number of cases we can determine the effect of singularities of $X$ on the degenerate intermediate Jacobian. This is joint work with S. Casalaina-Martin, S. Grushevsky and R. Laza.