The double cover of cubic surfaces branched along their Hessian

Abstract

We prove the relation between the Hodge structure of the double cover of a nonsingular cubic surface branched along its Hessian and the Hodge structure of the triple cover of the ambient projective space branched along the cubic surface. And we introduce a method to study the infinitesimal variations of Hodge structure of the double cover of the cubic surface. Using these results, we compute the N\'{e}ron-Severi lattices for the double cover of a generic cubic surface and the Fermat cubic surface.