Genus 2 curve configurations on Fano surfaces

TL;DR

This paper investigates the arrangements of genus 2 curves on Fano surfaces of cubic threefolds, linking automorphisms to these curves and classifying surfaces based on their automorphism groups.

Contribution

It provides a partial classification of Fano surfaces by automorphism groups and describes the configurations of genus 2 curves on them.

Findings

Automorphisms relate to genus 2 curve configurations.

Classification of Fano surfaces by automorphism groups.

The Klein cubic Fano surface has 55 genus 2 curves generating a rank 25 subgroup.

Abstract

We study the configurations of genus 2 curves on the Fano surfaces of cubic threefolds. We establish a link between some involutive automorphisms acting on such a surface S and genus 2 curves on S. We give a partial classification of the Fano surfaces according to the automorphism group generated by these involutions and determine the configurations of their genus 2 curves. We study the Fano surface of the Klein cubic threefold for which the 55 genus 2 curves generate a rank 25=h^{1,1} index 2 subgroup of the N\'eron-Severi group.