Change of polarization for moduli of sheaves on surfaces as Bridgeland wall-crossing

TL;DR

This paper demonstrates how certain geometric transformations of moduli spaces of sheaves on surfaces can be understood through Bridgeland wall-crossing, linking different stability conditions and polarization changes.

Contribution

It establishes a connection between Thaddeus flips and Bridgeland wall-crossing, providing a new perspective on moduli space transformations on surfaces.

Findings

Thaddeus flips can be realized via Bridgeland wall-crossing.

Change of polarization corresponds to varying stability conditions.

Provides a unified framework for understanding moduli space transformations.

Abstract

We prove that the "Thaddeus flips" of $L$-twisted sheaves constructed by Matsuki and Wentworth can be obtained via Bridgeland wall-crossing. Similarly, we realize the change of polarization for moduli spaces of 1-dimensional Gieseker semistable sheaves on a surface by varying a family of stability conditions.