On base point free theorem
[This paper was announced in Proc. Japan Acad., Ser. A 70, No. 6, 173-175 (1994; see the preceding review).] Let \(X\) be a non-singular projective variety with \(\dim X= n\) over \(\mathbb{C}\). And let \(\Delta = \sum^s_{i=1} \Delta_i\) be a reduced divisor on \(X\) with only simple normal crossings. Let \[ \begin{multlined}\text{Strata} (\Delta): =\{\Gamma \mid 1\leq k\leq n,\;1\leq i_1<i_2< \cdots< i_k\leq s,\\ \Gamma \text{ is an irreducible component of }\Delta_{i_1} \cap \Delta_{i_2} \cap \cdots \cap \Delta_{i_k} \neq \emptyset\}.\end{multlined} \] A divisor \(R\) on \((X,\Delta)\) is, by definition, nef and log big if \(R\) is nef and big and \(R|_\Gamma\) is nef and big for any member \(\Gamma\) of \(\text{Strata} (\Delta)\), due to \textit{M. Reid} [\S 10 in the English version of a paper by \textit{V. V. Shokurov}, Russ. Acad. Sci., Izv., Math. 40, No. 1, 95-202 (1993); translation from Izv. Ross. Akad. Nauk, Ser. Mat. 56, No. 1, 105-203 (1992; Zbl 0785.14023)]. Note that if \(R\) is ample then \(R\) is nef and log big on \((X, \Delta)\). -- 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 \(aL- (K_X+ \Delta)\) is nef and log big on \((X,\Delta)\) for some \(a\geq 0\). Then there exists a natural number \(l_1\), depending only on \(n\) and \(a\), such that the complete linear system \(|l_1L |\) is base point free.
scientific article; zbMATH DE number 949861