Notes on toric varieties from Mori theoretic viewpoint

Abstract

The main purpose of this notes is to supplement the paper by Reid: De- composition of toric morphisms, which treated Minimal Model Program (also called Mori's Program) on toric varieties. We compute lengths of negative extremal rays of toric varieties. As an application, a generalization of Fujita's c onjecture for singular toric varieties is obtained. We also prove that every toric variety has a small projective toric Q-factorialization. 0. Introduction. The main purpose of this notes is to supplement the paper by Reid (Re): Decomposition of toric morphisms, which treated Minimal Model Program (also called Mori's Program) on toric varieties. We compute lengths of negative extremal rays of toric varieties. This is an answer to (Ma, Remark-Question 10-3-6) for toric varieties, which is an easy exercise once we understand (Re). As a corollary, we obtain a strong version of Fujita's conjecture for singular toric varieties. Related topics are (Ft), (Ka), (La) and (Mu, Section 4). We will work, throughout this paper, over an algebraically closed field k of arbitrary characteristic.