Notes on Toric Varieties from Mori Theoretic Viewpoint

The main purpose of this notes is to supplement the paper [Re], which treated Minimal Model Program (also called Mori’s Program) on toric varieties. We calculate lengths of negative extremal rays of toric varieties. As an application, we obtain a generalization of Fujita’s conjecture for singular toric varieties. 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 [Re], which treated Minimal Model Program (also called Mori’s Program) on toric varieties. We calculate lengths of negative extremal rays of toric varieties. It 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], [L] and [Mu, Section 4]. We will freely use the notation in [Fl], [Re] and work over an algebraically closed field k of arbitrary characteristic throughout this paper. The following is the main theorem of this paper. Theorem 0.1 (Cone Theorem). Let X be an n-dimensional (not necessarily Q-factorial) projective toric variety over k. We can write the cone of curves as follows: