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 different from the blow-up . Here we give the complete classification of the corresponding pairs 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 différente de l'éclatement . Ici, on donne la classification complète des paires correspondantes 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.
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Toru Tsukioka 1
@article{CRMATH_2005__340_8_581_0, author = {Toru Tsukioka}, title = {Del {Pezzo} surface fibrations obtained by blow-up of a smooth curve in a projective manifold}, journal = {Comptes Rendus. Math\'ematique}, pages = {581--586}, publisher = {Elsevier}, volume = {340}, number = {8}, year = {2005}, doi = {10.1016/j.crma.2005.03.004}, language = {en}, }
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. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2005.03.004/
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