The Dimension of Quasi-Homogeneous Linear Systems With Multiplicity Four

TL;DR

This paper investigates the properties of quasi-homogeneous linear systems of plane curves with multiplicity four, providing partial proof of the Harbourne-Hirschowitz Conjecture for these systems.

Contribution

It proves the Harbourne-Hirschowitz Conjecture for linear systems with multiplicity four at all but one point, advancing understanding of special linear systems.

Findings

Proves the conjecture for multiplicity four cases with one exception

Characterizes when linear systems are special based on (-1) curves

Provides new insights into the structure of quasi-homogeneous systems

Abstract

A linear system of plane curves satisfying multiplicity conditions at points in general position is called special if the dimension is larger than the expected dimension. A (-1) curve is an irreducible curve with self intersection -1 and genus zero. The Harbourne-Hirschowitz Conjecture is that a linear system is special only if a multiple of some fixed (-1) curve is contained in every curve of the linear system. This conjecture is proven for linear systems with multiplicity four at all but one of the points.