Free linear systems on integral Gorenstein curves

Abstract

In the study of linear systems on smooth curves, it is enough to consider base point free linear systems. Indeed, each linear system is obtained by adding some base points to a uniquely determined base point free linear system. In [S], R. Hartshorne defined complete linear systems on integral Gorenstein curves. In [6, Example 1.6.11 he shows that the base point free linear systems are not that essential any more. In this paper, we introduce the notion of free linear systems on integral Gorenstein curves. We obtain that each linear system on such a curve is obtained by enlarging the base scheme (see Definition 2.1.2) of a uniquely determined free linear system. Singular points on the curve can appear as base points of free linear systems. We introduce the notion of base-point- degree of a linear system at a point. We prove that, in the case of free linear systems, this number is bounded from above by the kind of singularity. Moreover, let r be an integral curve on a smooth surface X and let P be a linear system on X. P induces a linear system on L’. We find upper bounds for the base-point-degree of this linear system in terms of intersection multiplicities. A very useful lemma in the study of linear systems on smooth curves is the base point free pencil trick (see, e.g., [ 1, p. 1261). In our context, we prove a free pencil trick. This is used in order to prove a result about free pencils on integral plane curves. In my paper [3], this result is essential for the study of linear systems on smooth plane curves. As an application; we show that, if C is the normalization of an integral plane curve f of degree d> 12 with 6 <