Linear systems of plane curves with a composite number of base points of equal multiplicity

Abstract:

In this article we study linear systems of plane curves of degree $d$ passing through general base points with the same multiplicity at each of them. These systems are known as homogeneous linear systems. We especially investigate for which of these systems, the base points, with their multiplicities, impose independent conditions and which homogeneous systems are empty. Such systems are called non-special. We extend the range of homogeneous linear systems that are known to be non-special. A theorem of Evain states that the systems of curves of degree $d$ with $4^h$ base points with equal multiplicity are non-special. The analogous result for $9^h$ points was conjectured. Both of these will follow, as corollaries, from the main theorem proved in this paper. Also, the case of $4^{h}9^{k}$ points will follow from our result. The proof uses a degeneration technique developed by C. Ciliberto and R. Miranda.