Special homogeneous linear systems on Hirzebruch surfaces

TL;DR

This paper verifies the Segre-Gimigliano-Harbourne-Hirschowitz Conjecture for Hirzebruch surfaces with equal base point multiplicities up to 8, advancing understanding of linear systems on these surfaces.

Contribution

It proves the conjecture for a new class of cases involving Hirzebruch surfaces with bounded multiplicities, extending previous results.

Findings

Conjecture holds for multiplicities ≤ 8 on Hirzebruch surfaces

Provides new evidence supporting the conjecture in algebraic geometry

Advances understanding of linear systems on Hirzebruch surfaces

Abstract

The Segre-Gimigliano-Harbourne-Hirschowitz Conjecture can be naturally formulated for Hirzebruch surfaces F_n. We show that this Conjecture holds for imposed base points of equal multiplicity bounded by 8.