Desingularization of some moduli scheme of stable sheaves on a surface

TL;DR

This paper constructs a desingularization of a moduli scheme of stable sheaves on a surface when crossing a wall in the stability space, and investigates the nature of singularities in specific cases.

Contribution

It provides a method to desingularize certain moduli schemes of stable sheaves using known smooth schemes, and analyzes singularities in special surface cases.

Findings

Successfully desingularized $M(H_+)$ using $M(H_-)$

Determined conditions under which singularities are terminal

Applied results to ruled and elliptic surfaces

Abstract

Let $X$ be a nonsingular projective surface over an algebraically closed field with characteristic zero, and $H_-$ and $H_+$ ample line bundles on $X$ separated by only one wall of type $(c_1,c_2)$. Suppose the moduli scheme $M(H_-)$ of rank-two $H_-$-stable sheaves with Chern classes $(c_1,c_2)$ is non-singular. We shall construct a desingularization of $M(H_+)$ by using $M(H_-)$. As an application, we consider whether singularities of $M(H_+)$ are terminal or not when $X$ is ruled or elliptic.