Base Point Freeness of Adjoint Linear Systems

Abstract

The purpose of this chapter is to discuss “base point freeness” of linear systems of adjoint type, i.e., linear systems of type $\left| {K_x + B} \right|,$, where K _X denotes the canonical divisor of a variety X and B is a (boundary) divisor with some specific conditions depending on the situation. As is clear from the formulation, the most natural framework for linear systems of adjoing type is that of the logarithmic category discussed in Chapter 2. Our key tool is the logarithmic version of the Kodaira vanishing theorem, i.e., the Kawamata—Viehweg vanishing theorem. Our viewpoint centering on adjoint linear systems, is in the spirit of Ein—Lazarsfeld [1], which applied the method of Kawamata—Reid—Shokurov to solve Fujita’s conjecture in dimension 3.