The Fano surface of the Klein cubic threefold

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 $11^{10}$.