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Let $X$ be a variety with at most terminal $\mathbb Q$-factorialsingularities 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 $\tau \geq 0$ is a rational number.Equivalently, $f$ is a Fano-Mori contraction associated to an extremal face in$\overline {NE(X)}_{K_X+\tau L = 0}$; these maps naturally arise in the contextof the minimal model program. We prove that, if $\tau > (n-3) >0$, the generalelement $X' \in |L|$ is a variety with at most terminal singularities. Then weapply this to characterize, via an inductive argument, some birationalcontractions as above with $\tau > (n-3)\geq 0$.

Local Fano-Mori contractions of high nef-value