Base Point Free Theorem of Reid-fukuda Type

Let (X,∆) be a proper dlt pair and L a nef Cartier divisor such that aL − (KX +∆) is nef and log big on (X,∆) for some a ∈ Z>0. Then |mL| is base point free for every m ≫ 0. 0. Introduction The purpose of this paper is to prove the following theorem. This type of base point freeness was suggested by M. Reid in [Re, 10.4]. Theorem 0.1 (Base point free theorem of Reid-Fukuda type). Assume that (X,∆) is a proper dlt pair. Let L be a nef Cartier divisor such that aL − (KX + ∆) is nef and log big on (X,∆) for some a ∈ Z>0. Then |mL| is base point free for every m ≫ 0, that is, there exists a positive integer m0 such that |mL| is base point free for every m ≥ m0. This theorem was proved by S. Fukuda in the case whereX is smooth and ∆ is a reduced simple normal crossing divisor in [Fk2]. In [Fk3], he proved it on the assumption that dimX ≤ 3 by using log Minimal Model Program. Our proof is similar to [Fk3]. However, we do not use log Minimal Model Program even in dimX ≤ 3. He also treated this problem under some extra conditions in [Fk4]. Acknowledgments . I would like to express my sincere gratitude to Dr. Daisuke Matsushita for giving me some comments. Notation. (1) We will make use of the standard notations and definitions as in [KoM]. (2) A pair (X,∆) denotes that X is a normal variety over C and ∆ is a Q-divisor on X such that KX +∆ is Q-Cartier. (3) Diff denotes the different (See [Utah, Chapter 16]). Date: February 1, 2008. 1991 Mathematics Subject Classification. 14C20. 1