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.

material is streamlined and some is moved to a forthcoming paper

Cited by in corpus (7)

• Discrete behavior of Seshadri constants on surfaces
• Deformation of canonical morphisms and the moduli of surfaces of general type
• Vanishing theorems for linearly obstructed divisors
• On linear systems of ${\mathbb{P}}^3$ with nine base points
• Collisions of fat points and applications to interpolation theory
• Lins Neto's examples of foliations and the Mori cone of blow-ups of $P^2$
• Linear Systems of Plane Curves with Base Points of Equal Multiplicity