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 $A=H-\sum {\epsilon }_i{E}_i$ be a generic polarization with $0<{\epsilon }_i\ll 1$. We classify the Chern characters which satisfy the weak Brill-Noether property, i.e. a general sheaf in $M_A(\mathbf{v})$, the moduli space of slope stable sheaves with Chern character $\mathbf{v}$, 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.