The double cover of cubic surfaces branched along their Hessian

TL;DR

This paper explores the Hodge structures of double covers of nonsingular cubic surfaces branched along their Hessians, establishing relations with triple covers and introducing methods to analyze their infinitesimal variations.

Contribution

It establishes a relation between the Hodge structures of double and triple covers of cubic surfaces and introduces a new method for studying their infinitesimal variations.

Findings

Computed Néron-Severi lattices for generic cubic surfaces

Analyzed Hodge structures of double covers

Established relations between double and triple covers

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.