  Let $X$ be the blow-up of $\mathbb{P}^2$ along $m$ general points, and
$A=H-\sum \varepsilon_iE_i$ be a generic polarization with
$0<\varepsilon_i\ll1$. We classify the Chern characters which satisfy the weak
Brill-Noether property, i.e. a general sheaf in $M_A({\bf{v}})$, the moduli
space of slope stable sheaves with Chern character ${\bf{v}}$, has at most one
non-zero cohomology. We further give a necessary and sufficient condition for
the existence of stable sheaves. Our strategy is to specialize to the case when
the $m$ points are collinear.
