Fano Manifolds with Long Extremal Rays

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 $\dim T \lt i_X$.

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Published 1 January 2005