ON CONTRACTIONS OF SMOOTH VARIETIES

Let phi : X - Z be a proper surjective map from a smooth complex manif old X onto a normal variety Z. If phi has connected fibers and - K-X i s phi-ample then phi is called a Fano-Mori contraction. In the present paper we study Fano-Mori contractions, fibers of which have dimension less than or equal to two: after describing possible two-dimensional isolated fibers me discuss their scheme-theoretic structure and the ge ometry of phi : X --> Z nearby such a fiber. If dimX = 4 and phi is bi rational with an isolated two-dimensional fiber then we obtain a compl ete description of phi. We provide also a description of a four-dimens ional conic fibration with an isolated fiber that is either a plane or a quadric. We construct pertinent examples.