Linear systems on rational surfaces

TL;DR

This paper proposes a conjecture about linear systems on Hirzebruch surfaces, inspired by the SHGH conjecture, and proves it for certain cases involving blow-ups at very general points.

Contribution

It introduces a new conjecture for Hirzebruch surfaces and proves its validity for specific blow-up configurations.

Findings

Conjecture holds for $F_e$ blown up at $r \,\leqslant \, e+4$ very general points.

Provides evidence supporting the conjecture in particular cases.

Advances understanding of linear systems on rational surfaces.

Abstract

Motivated by various equivalent versions of the SHGH conjecture for $\mathbb{P}^2$ blown up at very general points, we propose a similar conjecture for Hirzebruch surfaces. We prove that this conjecture is true for the Hirzebruch surface $\mathbb{F}_e$ blown up at $r\leqslant e+4$ very general points.