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Elliptic curves, fibrations and automorphisms of Fano surfaces. (Courbes elliptiques, fibrations et automorphismes des surfaces de Fano.) (French. English summary) Zbl 1191.14048
If $$S$$ 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.
##### MSC:
 14J30 $$3$$-folds 14J50 Automorphisms of surfaces and higher-dimensional varieties 14H52 Elliptic curves
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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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