The dimension of planar linear systems

Abstract

The space of algebraic curves satisfying multiplicity conditions at specified points is a linear system of plane curves, and is naturally a projective space. In some cases the conditions imposed are dependent, even when the points are chosen in general position, giving the system a larger dimension than is expected. Such linear systems are called special. The Harbourne-Hirschowitz Conjecture hypothesizes that a linear system will be special only if every curve of the system has a multiple of some fixed (-1) curve as a component. A linear system with a fixed multiplicity, m, assigned to all but one of the points is called a quasi-homogeneous linear system. The (-1) curves which may be contained in a quasi-homogeneous linear system are described herein. All linear systems with m = 4 which do contain a multiple (-1) curve are listed. A recent technique described by Rick Miranda and Ciro Ciliberto is then used to prove these are the only quasi-homogeneous linear systems with m = 4 which are special.