Local Fano–Mori contractions of high nef-value

Abstract

Let X be a variety with at most terminal Q-factorial singularities of dimension n. We study local contractions f : X ! Z supported by a Q- Cartier divisor of the type KX + �L, where L is an f-ample Cartier divisor and � � 0 is a rational number. Equivalently, f is a Fano-Mori contraction associated to an extremal face in NE(X) KX+�L=0 ; these maps naturally arise in the context of the minimal model program. We prove that, if � > (n 3) > 0, the general element X ' 2 |L| is a variety with at most terminal singularities. We apply this to characterize, via an inductive argument, some birational contractions as above with � > (n 3) � 0.