Moduli of framed sheaves on projective surfaces

TL;DR

This paper constructs a fine moduli space for torsion-free sheaves with a good framing on projective surfaces, showing it is a quasi-projective scheme and analyzing its smoothness and examples.

Contribution

It introduces a new moduli space for framed sheaves on surfaces, demonstrating its existence, structure, and properties, including stability and smoothness criteria.

Findings

Existence of a fine moduli space for framed sheaves

Characterization of smoothness obstructions

Examples on rational surfaces

Abstract

We show that there exists a fine moduli space for torsion-free sheaves on a projective surface, which have a "good framing" on a big and nef divisor. This moduli space is a quasi-projective scheme. This is accomplished by showing that such framed sheaves may be considered as stable pairs in the sense of Huybrechts and Lehn. We characterize the obstruction to the smoothness of the moduli space, and discuss some examples on rational surfaces.