Very ampleness of adjoint linear systems on smooth surfaces with boundary

Abstract

Let $M$ be a $\Q$-divisor on a smooth surface over $\C\,$. In this paper we give criteria for very ampleness of the adjoint of $\rup{M}$, the round-up of $M$. (Similar results for global generation were given by Ein and Lazarsfeld and used in their proof of Fujita's Conjecture in dimension 3.) In \S 4 we discuss an example which suggests that this kind of criteria might also be useful in the study of linear systems on surfaces.