The Fano surface of the Klein cubic threefold

TL;DR

This paper characterizes the unique automorphism properties of the Klein cubic threefold, computes its intermediate Jacobian's period lattice, and analyzes the geometric and algebraic structure of its Fano surface.

Contribution

It establishes the uniqueness of the Klein cubic threefold with an automorphism of order 11 and details the structure of its Fano surface and Néron-Severi group.

Findings

Klein cubic threefold is uniquely characterized by an automorphism of order 11.

Computed the period lattice of the intermediate Jacobian.

Determined the Néron-Severi group of the Fano surface, including rank and discriminant.

Abstract

We prove that the Klein cubic threefold $F$ is the only smooth cubic threefold which has an automorphism of order 11. We compute the period lattice of the intermediate Jacobian of $F$ and study its Fano surface $S$. We compute also the set of fibrations of $S$ onto a curve of positive genus and the intersection between the fibres of these fibrations. These fibres generate an index 2 sub-group of the N\'eron-Severi group and we obtain a set of generators of this group. The N\'eron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.