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

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 phi : X -> Z different from the blow-up pi : X -> Y. Here we give the complete classification of the corresponding pairs (Y, C) in the case where phi is a fiber type contraction of relative dimension 2, i.e. the general fibers of phi are del Pezzo surfaces. In Tsukioka (Thesis, Nancy University 1, 2005) the relative dimension I case is also considered. To cite this article: T Tsukioka, C. R. Acad. Sci. Paris, Ser. 1340 (2005). (c) 2005 Academie des sciences. Published by Elsevier SAS. All rights reserved.