  Let X be a projective variety with 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 \bar {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 rational number, and the fibres of f
have dimension less or equal to r+1.
