On base point free theorem

Abstract

IntroductionLet A" be a non-singular projective variety with dim X-n over C.And let Δ -Y ι \= ι Δ ι be a reduced divisor on X with only simple normal crossings.LetA divisor R on (X, Δ) is, by definition, nef and log big if R is nef and big and R\r is nef and big for any member Γ of Strata(J) (due to Reid [8, 10.4]).Note that if R is ample then R is nef and log big on (X, Δ).The purpose of this paper is to prove the following:MAIN THEOREM ("log effective freeness").Let L be a nef divisor on X such that aL-{K x -\-Δ) is nef and log big on {X, Δ) for some α^O.Then there exists a natural number l u depending only on n and a, such that the complete linear system \lχL\ is base point free.The paper is organized as follows.In Section 2 we recall known base point free theorems.In Section 3 we give a proof to Reid's "log eventual freedom" theorem in the "smooth" case.In Section 4 we prove the main theorem.In Section 5 we show some results concerning the log canonical divisor.We follow the notation and terminology of [Utah].The results of the present paper have been announced in [Resume].