Quasi-homogeneous linear systems on P2 with base points of multiplicity 7, 8, 9, 10

TL;DR

This paper proves the Harbourne-Hirschowitz conjecture for a class of quasi-homogeneous linear systems on the projective plane with specific multiplicities, advancing understanding of curve systems with multiple base points.

Contribution

It establishes the conjecture for systems with multiplicities 7, 8, 9, 10, filling a gap in the classification of such linear systems.

Findings

Confirmed the conjecture for multiplicities 7, 8, 9, 10

Characterized the structure of quasi-homogeneous systems on P2

Extended previous results to higher multiplicities

Abstract

In the paper we prove Harbourne-Hirschowitz conjecture for quasi-homogeneous linear systems on $\mathbb P^2$ for $m=7$, 8, 9, 10, i.e. systems of curves of given degree passing through points in general position with multiplicities at least $m,...,m,m_0$, where $m=7$, 8, 9, 10, $m_0$ is arbitrary.