On the second fundamental form of cubic threefolds [1]

Abstract : I will report on some geometric properties of the second fundamental form of the embedding which assigns to a cubic threefold X its intermediate Jacobian.The choice of a line on X defines a conic bundle structure on X over the projective plane. The discriminant curve is a quintic plane curve endowed with a natural etale double cover and the intermediate Jacobian is the Prym variety of this cover. We use the Hodge-Gaussian maps and the second Prym canonical Gaussian map associated with this quintic. This is a joint work with E. Colombo, J.C. Naranjo and G.P. Pirola.