The Cone of Curves of Algebraic Varieties

In this paper we announce a structure theorem on the cone of curves of algebraic varieties defined over a field of characteristic zero. Details will appear elsewhere. This theorem should be one of the key steps toward the theory of minimal models of algebraic varieties. We already have the so-called contraction theorem (Theorem 4), which is a generalization of Castelnuovo's criterion of exceptional curves of the. first kind. Our weak cone. theorem gua. rantees the. existence, of a good extremal ra. y to be contracted if the model is not minimal. The remaining thing to be. proved would be the theorem on elementa. rytra. nsformations (see Reid 5, 6, Kawamata 2). 1o We. fix our notation. Let X be a normal projective variety. We. define" N (X) 1-cycles on X / (R) R, N (X) line bundles on X / (R) Q, N (X) =N (X) (R) R, and NE (X) = the closed convex cone in N (X) generated by effective I-cycles, where denotes numerical equiva- lence. N (X) and N (X) are dual to each other by intersection pairing.