Degenerations of Planar Linear Systems

Abstract

Fixing $n$ general points $p_i$ in the plane, what is the dimension of the space of plane curves of degree $d$ having multiplicity $m_i$ at $p_i$ for each $i$? In this article we propose an approach to attack this problem, and demonstrate it by successfully computing this dimension for all $n$ and for $m_i$ constant, at most 3. This application, while previously known (see \cite{hirschowitz1}), demonstrates the utility of our approach, which is based on an analysis of the corresponding linear system on a degeneration of the plane itself, leading to a simple recursion for these dimensions. We also obtain results in the ``quasi-homogeneous'' case when all the multiplicities are equal except one; this is the natural family to consider in the recursion.