Journal of Mathematics of Kyoto University

The Fano surface of the Klein cubic threefold

Xavier Roulleau

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 Néron-Severi group and we obtain a set of generators of this group. The Néron-Severi group of $S$ has rank $25=h^{1,1}$ and discriminant $11^{10}$.

Article information

Source
J. Math. Kyoto Univ. Volume 49, Number 1 (2009), 113-129.

Dates
First available in Project Euclid: 30 July 2009

https://projecteuclid.org/euclid.kjm/1248983032

Digital Object Identifier
doi:10.1215/kjm/1248983032

Mathematical Reviews number (MathSciNet)
MR2531132

Zentralblatt MATH identifier
1207.14045

Citation

Roulleau, Xavier. The Fano surface of the Klein cubic threefold. J. Math. Kyoto Univ. 49 (2009), no. 1, 113--129. doi:10.1215/kjm/1248983032. https://projecteuclid.org/euclid.kjm/1248983032