Fano surfaces with 12 or 30 elliptic curves

TL;DR

This paper investigates Fano surfaces associated with cubic threefolds containing 12 or 30 elliptic curves, determining their Picard number and Néron-Severi group structure, revealing new geometric properties.

Contribution

It provides a detailed analysis of the Picard number and Néron-Severi group basis for Fano surfaces with specific elliptic curve configurations, including the Fermat cubic case.

Findings

Determined Picard numbers for Fano surfaces with 12 or 30 elliptic curves.

Computed a basis for the Néron-Severi group of the Fano surface of the Fermat cubic.

Identified geometric properties related to low genus curves on these surfaces.

Abstract

Curves of low genus on a surface carry important informations on that surface. We study the Fano surfaces of lines of cubic threefolds that contain 12 or 30 elliptic curves. We determine their Picard number and compute a basis of the N\'eron-Severi group of the Fano surface of the Fermat cubic threefold.