The second fundamental form on the moduli space of cubic threefolds in A5

Abstract

We study the second fundamental form of the Siegel metric in A5 restricted to the locus of intermediate Jacobians of cubic threefolds. We prove that the image of this second fundamental form, which is known to be non-trivial, is contained in the kernel of a suitable multiplication map. Some ingredients are: the conic bundle structure of cubic threefolds, Prym theory, Gaussian maps and Jacobian ideals. © by the author(s)