Elliptic curves, fibrations and automorphisms of Fano surfaces.
(Courbes elliptiques, fibrations et automorphismes des surfaces de Fano.)

*(French. English summary)*Zbl 1191.14048If \(S\) is a smooth surface of general type, such that \(\Omega_S\) is generated by global sections and the irregularity \(q\) is at least \(4\), it is possible to define in a natural way the cotangent map \(\psi:\mathbb P(T_S)\to\mathbb P^{q-1}\). The non-ample curves of \(S\) are then the irreducible components of dimension one of the image into \(S\) of \(\psi^{-1}(p)\), for the points \(p\) such that this fibre has positive dimension.

The author consider the case in which \(S\) is the Fano surface of the lines on a cubic threefold \(F\subset \mathbb P^4\), in which case the image of \(\psi\) is isomorphic to \(F\). He claims that the non-ample curves are the smooth elliptic curves. To such a curve \(E\), he associates an involution \(\sigma_E\) of \(S\) (fixing \(E\) and \(27\) isolated points), a rational map \(\gamma_E\) from \(S\) to \(E\), and a reflection \(M_{\sigma_E}\in \mathrm{GL}(H^0(\Omega_S)^*)\).

The author gives the list of the groups generated by these reflections, and in each case computes the number of elliptic curves on \(S\). The Fermat’s cubic is the only cubic hypersurface for which \(S\) contains the maximum number, \(30\), of elliptic curves. In this case the Albanese variety of \(S\) and the configuration of the elliptic curves are described. Finally, the author considers the case of the Klein’s cubic threefold.

Reviewer’s remark: The article contains some examples but no proofs.

The author consider the case in which \(S\) is the Fano surface of the lines on a cubic threefold \(F\subset \mathbb P^4\), in which case the image of \(\psi\) is isomorphic to \(F\). He claims that the non-ample curves are the smooth elliptic curves. To such a curve \(E\), he associates an involution \(\sigma_E\) of \(S\) (fixing \(E\) and \(27\) isolated points), a rational map \(\gamma_E\) from \(S\) to \(E\), and a reflection \(M_{\sigma_E}\in \mathrm{GL}(H^0(\Omega_S)^*)\).

The author gives the list of the groups generated by these reflections, and in each case computes the number of elliptic curves on \(S\). The Fermat’s cubic is the only cubic hypersurface for which \(S\) contains the maximum number, \(30\), of elliptic curves. In this case the Albanese variety of \(S\) and the configuration of the elliptic curves are described. Finally, the author considers the case of the Klein’s cubic threefold.

Reviewer’s remark: The article contains some examples but no proofs.

Reviewer: Emilia Mezzetti (Trieste)

##### MSC:

14J30 | \(3\)-folds |

14J50 | Automorphisms of surfaces and higher-dimensional varieties |

14H52 | Elliptic curves |

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\textit{X. Roulleau}, C. R., Math., Acad. Sci. Paris 345, No. 4, 209--212 (2007; Zbl 1191.14048)

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##### References:

[1] | Clemens, H.; Griffiths, P., The intermediate Jacobian of the cubic threefold, Ann. of math., 95, 281-356, (1972) · Zbl 0214.48302 |

[2] | Hirzebruch, F., Arrangements of lines and algebraic surfaces, (), 113-140 |

[3] | X. Roulleau, L’application cotangente des surfaces de type général, Thèse, Université d’Angers, 2007 |

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