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 of <(NE(X))over bar> such that R center dot(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 center dot(K-X+rL)< 0, where r is a non-negative integer, and the fibres of f have dimension less than or equal to r+1.