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

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