On the extendability of elliptic surfaces of rank two and higher

Lopez, Angelo Felice; Muñoz, Roberto; Sierra, José Carlos

Annales de l'Institut Fourier, Tome 59 (2009) no. 1, pp. 311-346.

On étudie les variétés de dimension trois $X\subset {\mathbb{P}}^r$ qui ont comme section hyperplane une surface lisse avec une fibration elliptique. On prouve d’abord un théorème général sur les plongements possibles de ces surfaces de nombre de Picard égal à deux. Dans un deuxième temps, on prouve des résultats plus précis pour les fibrations de Weierstrass de rang supérieur ou égal à deux. En particulier, on prouve qu’une fibration de Weierstrass de rang deux qui n’est pas une surface K3 n’est pas une section hyperplane d’une variété de dimension trois localement intersection complète. On donne, de plus, des conditions sous lesquelles, pour beaucoup des plongements de fibrations de Weierstrass de rang quelconque, toute variété de dimension trois comme ci-dessus est un cône.

We study threefolds $X\subset {\mathbb{P}}^r$ having as hyperplane section a smooth surface with an elliptic fibration. We first give a general theorem about the possible embeddings of such surfaces with Picard number two. More precise results are then proved for Weierstrass fibrations, both of rank two and higher. In particular we prove that a Weierstrass fibration of rank two that is not a K3 surface is not hyperplane section of a locally complete intersection threefold and we give some conditions, for many embeddings of Weierstrass fibrations of any rank, under which every such threefold must be a cone.

Classification : 14J30, 14J27, 11G05
Keywords: Elliptic surfaces, hyperplane sections, Mori fiber spaces
Mot clés : surfaces elliptiques, sections hyperplanes, fibration de Mori

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