Reduction method for linear systems of plane curves with base fat points

TL;DR

This paper introduces a new method to prove the non-speciality of linear systems of plane curves with base fat points, confirming the Hirschowitz-Harbourne Conjecture for points with multiplicity up to 42.

Contribution

A novel proof technique for non-speciality of linear systems, verifying the conjecture for a broad class of base points with bounded multiplicity.

Findings

Hirschowitz-Harbourne Conjecture verified for multiplicity ≤ 42

New method simplifies proving non-speciality

Applicable to systems with base points in general position

Abstract

In the paper we develop a new method of proving non-speciality of a linear system with base fat points in general position. Using this method we show that the Hirschowitz-Harbourne Conjecture holds for systems with base points of equal multiplicity bounded by 42.