The Fano surface of the Klein cubic threefold.

*(English)*Zbl 1207.14045The set of lines on a smooth cubic threefold \(F\) is parametrized by the so-called Fano surface of the threefold, a surface embedded in the intermediate Jacobian of \(F\) by its own Albanese morphism.

The Tangent Bundle Theorem enables to recover the cubic from its Fano surface. The author had previousy proved [Manuscr. Math. 129, No. 3, 381–399 (2009; Zbl 1177.14079)] that the automorphism group of the threefold is isomorphic to the automorphism group of its Fano surface.

In this paper he pursue the study of these groups. He first proves that the order of this group divides \(11\cdot 7 \cdot 5^2 \cdot 3^{6} \cdot 2^{23}\). Then he shows that a smooth cubic threefold cannot have an automorphism of order \(7\), and the the only cubic threefold with an automorphism of order \(11\) is the Klein cubic \[ x_1x_5^2+x_5x_3^2+x_3x_4^2+x_4x_2^2+x_2x_1^2=0. \] The automorphism group of the Klein cubic is isomorphic to \(PSL(2,11)\), as shown by A. Adler [J. Algebra 72, 146–165 (1981; Zbl 0479.20020)]. Using that, the author computes the period lattice of the Fano surface of the Klein cubic, the rank and the discriminant of its Néron-Severi group. He computes also the numerical classes of all connected fibrations of the Fano surface onto a curve of positive genus.

The Tangent Bundle Theorem enables to recover the cubic from its Fano surface. The author had previousy proved [Manuscr. Math. 129, No. 3, 381–399 (2009; Zbl 1177.14079)] that the automorphism group of the threefold is isomorphic to the automorphism group of its Fano surface.

In this paper he pursue the study of these groups. He first proves that the order of this group divides \(11\cdot 7 \cdot 5^2 \cdot 3^{6} \cdot 2^{23}\). Then he shows that a smooth cubic threefold cannot have an automorphism of order \(7\), and the the only cubic threefold with an automorphism of order \(11\) is the Klein cubic \[ x_1x_5^2+x_5x_3^2+x_3x_4^2+x_4x_2^2+x_2x_1^2=0. \] The automorphism group of the Klein cubic is isomorphic to \(PSL(2,11)\), as shown by A. Adler [J. Algebra 72, 146–165 (1981; Zbl 0479.20020)]. Using that, the author computes the period lattice of the Fano surface of the Klein cubic, the rank and the discriminant of its Néron-Severi group. He computes also the numerical classes of all connected fibrations of the Fano surface onto a curve of positive genus.

Reviewer: Roberto Pignatelli (Trento)