Fano-Mori contractions of high length on projective varieties with terminal singularities

Let $X$ be a projective variety with $\Q$-factorial terminal singularities and let $L$ be an ample Cartier divisor on $X$. We prove that if $f$ is a birational contraction associated to an extremal ray $ R \subset \overline {NE(X)}$ such that $R.(K_X+(n-2)L)<0$ then $f$ is a weighted blow-up of a smooth point. We then classify divisorial contractions associated to extremal rays $R$ such that $R.(K_X+rL)<0$, where $r$ is a non-negative integer, and the fibres of $f$ have dimension less or equal to $r+1$.