DEGENERATIONS OF PLANAR LINEAR-SYSTEMS

Fixing n general points p(i) in the plane, what is the dimension of th e space of plane curves of degree cl having multiplicity m(i) at p(i) for each i? In this article we propose an approach to attack this prob lem, and demonstrate it by successfully computing this dimension for a ll n and for m(i) constant, at most 3. This application, while previou sly known (see [Hi]), demonstrates the utility of our approach, which is based on an analysis of the corresponding linear system on a degene ration of the plane itself, leading to a simple recursion for these di mensions. We also obtain results in the ''quasi-homogeneous'' case whe n all the multiplicities are equal except one; this is the natural fam ily to consider in the recursion.