A new proof of the nonrationality of cubic threefolds

Abstract

A new proof of the non-rationality of a generic cubic threefold is given as follows: If a generic cubic threefold were rational then the associated intermediate Jacobian would be a product of Jacobians of curves. We degenerate a generic cubic threefold to the Segre Cubic Threefold and so there is a degeneration of intermediate Jacobians as well. Associated to the degenerating family of Pryms is a unimodular system of vectors. Rationality of the generic cubic threefold would imply that the unimodular system would be cographic dicing. However, we show that the unimodular system obtained is a well known symmetric non-cographic dicing called E_5.