Ample vector bundles with sections vanishing on special varieties

Abstract

Let E be an ample vector bundle of rank r on a complex projective manifold X such that there exists a section $s \in Γ(\cal E)$ whose zero locus Z = (s = 0) is a smooth submanifold of the expected dimension dim X - r: = n -r. Assume that Z is not minimal; we investigate the hypothesis under which the extremal contractions of Z can be lifted to X. Finally we study in detail the cases in which Z is a scroll, a quadric bundle or a del Pezzo fibration.