Vector Bundles of Rank 2 and Linear Systems on Algebraic Surfaces

Abstract

Let S be a smooth complex algebraic surface and let L be a divisor on S. It is well-known that the linear system IL + KsI, where Ks is the canonical divisor, plays an important role in understanding the geometry of S. Several authors (see, for example, [8], [11]) studied various properties of this system. Our paper treats the same subject, but the point of view is different from the works mentioned above. The philosophy underlying our approach is almost classical: Points on S (more generally, effective 0-cycles) in special position with respect to IL + KsI (see definition below) contain information about the geometry of S. This point of view was recently revived in [5] (also [10]). In fact, this paper contains the technique of going from the effective 0-cycle Z in special position with respect to IL + KsI to the geometry of S. For the sake of completeness we give a definition and state a theorem which can be found in [5], [10].