Global generation of pluricanonical and adjoint linear series on smooth projective threefolds

Abstract

The purpose of this paper is to show how the cohomological techniques developed by Kawamata, Reid, Shokurov, and others lead to some effective and practical results of Reider-type on freeness of linear series on smooth complex projective threefolds. In recent years, there has been a great deal of interest in the geometric properties of pluricanonical and adjoint linear systems on surfaces and higherdimensional varieties. Among other things, one wants to understand as explicitly as possible when the systems in question are base-point free or very ample. Modem work in this area goes back to Kodaira [Kod] and Bombieri [Bmb], who studied pluricanonical maps of surfaces of general type. More recently, many of their results have emerged as special cases of a celebrated theorem of Reider [Rdr]. Reider uses vector bundle techniques to show that, if B is a nef line bundle on a surface X such that B2 > 5, then KX + B is globally generated unless there exist certain special curves C c X such that B * C < 1; he also obtains analogous criteria for KX + B to be very ample. A cohomological approach to these theorems, based on Miyaoka's vanishing theorem for Zariski decompositions, was given by Sakai [Sak2].