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

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 $h^1(F_e,E(tM))=0$.

7 pages, no figures, to appear on Cent. Eur. J. Math