Fano manifolds with long extremal rays

Abstract

Let X be a Fano manifold of pseudoindex i_X whose Picard number is at least two and let R be an extremal ray of X with exceptional locus Exc(R). We prove an inequality which bounds the length of R in terms of i_X and of the dimension of Exc(R) and we investigate the border cases. In particular we classify Fano manifolds X of pseudoindex i_X obtained blowing up a smooth variety Y along a smooth subvariety T such that dim T < i_X.