The Fano surface of the Fermat cubic threefold, the del Pezzo surface of degree 5 and a ball quotient

TL;DR

This paper investigates the Fano surface of the Fermat cubic threefold, revealing its structure as an abelian cover of a del Pezzo surface and its relation to a ball quotient lattice, connecting algebraic geometry and lattice theory.

Contribution

It establishes the Fano surface as a degree 81 abelian cover of a degree 5 del Pezzo surface and links its complement to a specific ball quotient lattice.

Findings

Fano surface is a degree 81 abelian cover of the del Pezzo surface.

Complement of 12 elliptic curves forms a ball quotient.

The lattice is related to the Deligne-Mostow lattice number 1.

Abstract

We study the Fano surface S of the Fermat cubic threefold. We prove that S is a degree 81 abelian cover of the degree 5 del Pezzo surface and that the complement of the union of 12 disjoint elliptic curves on S is a ball quotient. The lattice of this ball quotient is related to the Deligne-Mostow lattice number 1.