Cone Theorem Revisited

Abstract

The purpose of this chapter is to present Mori’s original idea (cf. Mori [1][2]) to prove the cone theorem (in the smooth case) through his ingenious method of “bend and break” to produce rational curves of some bounded degree (with respect to an ample divisor or to the canonical divisor). This method leads to the result of Miyaoka—Mori [1] claiming the uniruledness of Mori fiber spaces, yielding the generalization by Kawamata [13] claiming that (every irreducible component of) the exceptional locus of an extremal contraction is also uniruled in general.