Elliptic curve configurations on Fano surfaces

TL;DR

This paper classifies the configurations of elliptic curves on Fano surfaces of smooth cubic threefolds, detailing their counts, intersections, and models, and relates these configurations to automorphism group classifications.

Contribution

It provides a complete classification of elliptic curve configurations on Fano surfaces, including their counts, intersections, and models, and links these to automorphism groups.

Findings

Number of elliptic curves on Fano surfaces determined

Configurations characterized by intersections and plane models

Connections established between elliptic configurations and automorphism groups

Abstract

The elliptic curves on a surface of general type constitute an obstruction for the cotangent sheaf to be ample. In this paper, we give the classification of the configurations of the elliptic curves on the Fano surface of a smooth cubic threefold. That means that we give the number of such curves, their intersections and a plane model. This classification is linked to the classification of the automorphism groups of theses surfaces.