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

TL;DR

This paper explicitly describes higher direct images of the structure sheaf via the Hilbert-Chow morphism for certain moduli spaces on smooth projective surfaces, providing formulas and confirming conjectures in specific cases.

Contribution

It provides explicit descriptions of higher direct images of the structure sheaf under the Hilbert-Chow morphism for moduli spaces of sheaves on surfaces, including formulas for Euler characteristics.

Findings

Higher direct images are direct sums of line bundles.

Explicit formulas for Euler characteristics in the case of .

Confirmation of a conjecture by Chung-Moon for .

Abstract

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 $\pi:M\rightarrow |L|$. We give explicit forms of the higher direct images $R^i\pi_*\mathcal{O}_M$ under some mild conditions on $M$ and $|L|$. Our result shows that $R^i\pi_*\mathcal{O}_M$ are direct sums of line bundles. In particular, using our result one gets explicit formulas for the Euler characteristic of $\pi^*\mathcal{O}_{|L|}(m)$, which in $X=\mathbb{P}^2$ case was once conjectured by Chung-Moon.