Suppose $\xi$ is chern character on $\mathbb P^2$. Then there is a moduli space $M(\xi)$ of semistable sheaves of chern character $\xi$. 


If $\xi$ has Euler characteristic 0, then apparently there is a theorem of Hirschowitz and Gottsche that general sheaf in the moduli space has no cohomology.


But now suppose that $\xi$ has positive Euler characteristic. Is is true that a general sheaf has no $H^1$?


If $\xi$ has negative Euler characteristic, does a general sheaf have only $H^1$ (i.e. $H^0$ and $H^2$ vanish)?



Do general sheaves on P^2 have cohomology governed by their Euler characteristic?
See Le Potier's Lectures on Vector Bundles, p 232, Theorem 18.1.1, where he quotes Gottsche and Hirschowitz. Part (ii) is exactly what was requested. The hypothesis needed is $\mu>-3$. (The example $I\_p(-3)$ shows that some hypothesis like this is necessary.) 


