Local Fano-Mori contractions of high nef-value

Let $X$ be a variety with at most terminal $\mathbb{Q}$-factorial 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 $\tau \ge 0$ 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 context of the minimal model program. We prove that, if $\tau >(n-3)>0$, the general element $X^{\mathrm{\prime }}\in \mathrm{\mid }L\mathrm{\mid }$ is a variety with at most terminal singularities. Then we apply this to characterize, via an inductive argument, some birational contractions as above with $\tau >(n-3)\ge 0$.