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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 weakBrill-Noether property, i.e. a general sheaf in $M_A({\bf{v}})$, the modulispace of slope stable sheaves with Chern character ${\bf{v}}$, has at most onenon-zero cohomology. We further give a necessary and sufficient condition forthe existence of stable sheaves. Our strategy is to specialize to the case whenthe $m$ points are collinear.

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