Higher direct images of the structure sheaf via the Hilbert-Chow morphism

Let $X$ be a projective smooth surface over $\mathbb{C}$ with $H^2({\mathcal{O}}_X)=0$. Let $M=M(L,\chi )$ be the moduli space of 1-dimensional semistable sheaves with determinant ${\mathcal{O}}_X(L)$ and Euler characteristic $\chi$. We have the Hilbert-Chow morphism $\pi :M\to \mathrm{\mid }L\mathrm{\mid }$. We give explicit forms of the higher direct images $R^i{\pi }_{\ast }{\mathcal{O}}_M$ under some mild conditions on $M$ and $\mathrm{\mid }L\mathrm{\mid }$. Our result shows that $R^i{\pi }_{\ast }{\mathcal{O}}_M$ are direct sums of line bundles. In particular, using our result one gets explicit formulas for the Euler characteristic of ${\pi }^{\ast }{\mathcal{O}}_{\mathrm{\mid }L\mathrm{\mid }}(m)$, which in $X={\mathbb{P}}^2$ case was once conjectured by Chung-Moon.