SOME REMARKS ON THE STUDY OF GOOD CONTRACTIONS

Let X be a complex projective variety with log terminal singularities admitting an extremal contraction in terms of Minimal Model Theory, i.e. a projective morphism phi : X --> Z onto a normal variety Z with connected fibers which is given by a (high multiple of a) divisor of the type K-X + rL, where r is a positive rational number and L is an ample Cartier divisor. We first prove that the dimension of any fiber F of phi is bigger or equal to (r - 1) and, if phi is birational, that dimF greater than or equal to r, with the equalities if and only if F is the projective space and L the hyperplane bundle (this is a sort of ''relative'' version of a theorem of Kobayashi-Ochiai). Then we describe the structure of the morphism phi itself in the case in which all fibers have minimal dimension with the respect to r. If phi is a birational divisorial contraction and X has terminal singularities we prove that phi is actually a ''blow-up''.