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 → Z supported by a ℚ-Cartier divisor of the type KX + τL, where L is an f-ample Cartier divisor and τ > 0 is a rational number. Equivalently, f is a Fano-Mori contraction associated to an extremal face in NE(X)KX+τL=0. We prove that, if τ > (n - 3) > 0, the general element X′ ϵ \L\ is a variety with at most terminal singularities. We apply this to characterize, via an inductive argument, some birational contractions as above with τ > (n - 3) > 0.