Moduli of semistable sheaves as quiver moduli

TL;DR

This paper explores how moduli spaces of semistable sheaves on projective surfaces can be described as quiver moduli spaces, using derived categories and exceptional collections, extending previous results on ^2.

Contribution

It provides a new perspective on moduli spaces via derived categories and extends the quiver description to ^1 ^1 surfaces.

Findings

Moduli spaces of Gieseker-semistable sheaves can be realized as quiver moduli spaces.

The approach uses t-structures and exceptional collections in derived categories.

Extension of the quiver moduli description to ^1 ^1 surfaces.

Abstract

In the 1980s Dr\'ezet and Le Potier realized moduli spaces of Gieseker-semistable sheaves on $\mathbb{P}^2$ as what are now called quiver moduli spaces. We discuss how this construction can be understood using t-structures and exceptional collections on derived categories, and how it can be extended to a similar result on $\mathbb{P}^1\times\mathbb{P}^1$.