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 $\operatorname{Exc}(R)$. We prove an inequality which bounds the length of $R$ in terms of $i_X$ and of the dimension of $\operatorname{Exc}(R)$ 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 $\dim T < i_X$.