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

Algebraic Geometry