On the local geometry of the moduli space of (2,2)-threefolds in A9

Abstract

We study the local geometry of the moduli space of intermediate Jacobians of (2,2)-threefolds in P2× P2. More precisely, we prove that a composition of the second fundamental form of the Siegel metric in A9 restricted to this moduli space, with a natural multiplication map is a nonzero holomorphic section of a vector bundle. We also describe its kernel. We use the two conic bundle structures of these threefolds, Prym theory, gaussian maps and Jacobian ideals.