Moduli Spaces of Sheaves on General Blow-ups of $\mathbb{P}^2$

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.