Vector bundles on Hirzebruch surfaces whose twists by a non-ample line bundle have natural cohomology

TL;DR

This paper investigates vector bundles on Hirzebruch surfaces that exhibit natural cohomology when twisted by a specific non-ample line bundle, revealing conditions under which cohomology behaves predictably.

Contribution

It characterizes vector bundles on Hirzebruch surfaces with natural cohomology upon twisting by a non-ample, spanned line bundle, extending understanding of cohomological behavior in this context.

Findings

Identifies conditions for natural cohomology in twisted vector bundles

Provides criteria for cohomology vanishing and non-vanishing

Enhances understanding of vector bundle behavior on Hirzebruch surfaces

Abstract

Here we study vector bundles $E$ on the Hirzebruch surface $F_e$ such that their twists by a spanned, but not ample, line bundle $M = \mathcal {O}_{F_e}(h+ef)$ have natural cohomology, i.e. $h^0(F_e,E(tM)) >0$ implies $h^1(F_e,E(tM)) = 0$.