ar X iv : a lg - g eo m / 9 61 10 17 v 1 1 4 N ov 1 99 6 A VIEW ON CONTRACTIONS OF HIGHER DIMENSIONAL VARIETIES

Introduction. In this paper we discuss some recent results about maps of complex algebraic varieties. A contraction ϕ : X → Z is a proper surjective map of normal varieties with connected fibers. We will always assume that X, if not smooth, has mild singularities; in particular we assume that a rational multiple of the canonical sheaf, rK X , is a Cartier divisor. In this hypothesis the contraction ϕ is called Fano-Mori, or extremal, or just good if the anticanonical divisor of X, denoted by −K X , is ϕ-ample. Fano-Mori contractions occur naturally in classification theories of algebraic varieties starting with the theory of surfaces. This becomes apparent from the following list of contractions of smooth surfaces. Namely, suppose that ϕ : X → Z is an extremal contraction of a smooth surface X, then one of the following occurs: (a) Z is a point and X is a del Pezzo surface; (b) Z is a smooth curve and ϕ : X → Z is a conic or P 1-bundle, in particular every fiber of ϕ is reduced and isomorphic to P 1 or to a union of two P 1 's meeting transversally; (c) Z is a smooth surface (thus ϕ is birational) and the exceptional locus consists of disjoint smooth rational curves with normal bundle O(−1), thus ϕ is a composition of blow-downs of disjoint rational curves to smooth points on Z. This is a classical result due to the Italian school of algebraic geometry; for a modern presentation see for instance [B-P-V]. As it follows from the 2-dimensional example, contractions can be either birational or of fiber type if dimZ < dimX. A birational contraction ϕ is called crepant if K X = ϕ * K Z. In case of surfaces crepant contractions lead to Du Val singularities which are also called simple rational double points or A–D–E-singularities (see [B-P-V]). A classical application of Fano-Mori contractions is the general adjunction theory as presented in [B-S3]. If L is an ample line bundle on X and K X is not nef then one considers an adjoint divisor K X + τ L, where τ > 0 is a rational number such