Del Pezzo surface fibrations obtained by blow-up of a smooth curve in a projective manifold

Toru Tsukioka

doi:10.1016/j.crma.2005.03.004

Algebraic Geometry

Del Pezzo surface fibrations obtained by blow-up of a smooth curve in a projective manifold

Presented by: Jean-Pierre Demailly

Toru Tsukioka ¹

¹ Institut Élie-Cartan, université H. Poincaré, Nancy 1, UMR 7502, BP 239, 54506 Vandoeuvre-lès-Nancy cedex, France

Comptes Rendus. Mathématique, Volume 340 (2005) no. 8, pp. 581-586

Abstract (Original language)
Résumé

This Note is devoted to the study of the Fano manifolds X obtained by blow-up along a smooth curve C in a complex projective manifold Y. By the Mori theory, we can ensure the existence of an extremal contraction $φ : X \to Z$ different from the blow-up $π : X \to Y$ . Here we give the complete classification of the corresponding pairs $(Y, C)$ in the case where φ is a fiber type contraction of relative dimension 2, i.e. the general fibers of φ are del Pezzo surfaces. In Tsukioka (Thesis, Nancy University 1, 2005), the relative dimension 1 case is also considered.

Cette Note est consacrée à l'étude des variétés de Fano X obtenues par éclatement d'une courbe lisse C dans une variété projective complexe et lisse Y. D'après la théorie de Mori, on peut assurer l'existence d'une contraction extrémale $φ : X \to Z$ différente de l'éclatement $π : X \to Y$ . Ici, on donne la classification complète des paires correspondantes $(Y, C)$ dans le cas où φ est de type fibrant de dimension relative 2, c'est-à-dire quand les fibres générales de φ sont des surfaces de del Pezzo. Dans Tsukioka (thèse, université Nancy 1, 2005) le cas de dimension relative 1 est aussi étudié.

Received: 2005-01-27
Accepted: 2005-03-04
Published online: 2005-04-07

DOI: 10.1016/j.crma.2005.03.004

Author's affiliations:

Toru Tsukioka ¹

¹ Institut Élie-Cartan, université H. Poincaré, Nancy 1, UMR 7502, BP 239, 54506 Vandoeuvre-lès-Nancy cedex, France

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     author = {Toru Tsukioka},
     title = {Del {Pezzo} surface fibrations obtained by blow-up of a smooth curve in a projective manifold},
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     doi = {10.1016/j.crma.2005.03.004},
     language = {en},
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Toru Tsukioka. Del Pezzo surface fibrations obtained by blow-up of a smooth curve in a projective manifold. Comptes Rendus. Mathématique, Volume 340 (2005) no. 8, pp. 581-586. doi: 10.1016/j.crma.2005.03.004

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