Journal of Mathematics of Kyoto University

The Fano surface of the Klein cubic threefold

Xavier Roulleau

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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

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

First available in Project Euclid: 30 July 2009

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Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 14J29: Surfaces of general type 14J50: Automorphisms of surfaces and higher-dimensional varieties
Secondary: 14J70: Hypersurfaces 32G20: Period matrices, variation of Hodge structure; degenerations [See also 14D05, 14D07, 14K30]


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.

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