A note on Fano surfaces of nodal cubic threefolds

TL;DR

This paper investigates the Picard variety of Fano surfaces associated with nodal and mildly cuspidal cubic threefolds, establishing connections between divisors on these surfaces and those on symmetric products of genus 4 curves.

Contribution

It introduces a novel approach linking divisors on Fano surfaces to symmetric products of genus 4 curves, expanding understanding of their Picard varieties in various characteristics.

Findings

Established relations between divisors on Fano surfaces and symmetric products of genus 4 curves

Extended analysis to arbitrary characteristic fields

Provided new insights into the Picard variety structure of these Fano surfaces

Abstract

We study the Picard variety of the Fano surface of nodal and mildly cuspidal cubic threefolds in arbitrary characteristic by relating divisors on the Fano surface to divisors on the symmetric product of a curve of genus 4.