Local Fano–Mori contractions of high nef-value

Let $X$ be a variety with terminal singularities of dimension $n$. We study local contractions $f : X \to Z$ supported by a $\mathbb{Q}$-Cartier divisor of the type $K_X + \tau L$, where $L$ is an $f$-ample Cartier divisor and $\tau \gt 0$ is a rational number. Equivalently, $f$ is a Fano–Mori contraction associated to an extremal face in $\overline{NE(X)}_{KX + \tau L=0}$. We prove that, if $\tau \gt (n - 3) \gt 0$, the general element $X^{\prime} \in \lvert L \rvert$ is a variety with at most terminal singularities. We apply this to characterize, via an inductive argument, some birational contractions as above with $\tau \gt (n - 3) \geq 0$.

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Accepted 14 April 2015

Published 12 January 2017