On the cone of curves of algebraic varieties
Yūjirō Kawamata · Proceedings of the Japan Academy Series A Mathematical Sciences · 1983
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.