# The Fano surface of the Klein cubic threefold - Mathematics > Algebraic Geometry

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

Author: ** Xavier Roulleau**

Source: https://arxiv.org/