Cubic threefolds have been studied in algebraic geometry since classical times. In [5], Clemens and Griffiths proved that the intermediate Jacobian JX of a smooth cubic threefold is not isomorphic as a principally polarized abelian variety to a product of Jacobians of curves, which implies the irrationality of X. They also established the Torelli theorem for cubic threefolds: the principally po...

We characterize the class of integral square matrices M having the property that for every integral vector q the linear complementarity problem with data M; q has only integral basic solutions. These matrices, called principally unimodular matrices, are those for which every principal nonsingular submatrix is unimodular. As a consequence , we show that if M is rank-symmetric and principally uni...

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 ha...