# Bridgeland wall-crossing for some Simpson moduli spaces on $\mathbb{P}^1 \times \mathbb{P}^1$

## TL;DR

This paper studies how Bridgeland stability conditions cause wall-crossing phenomena in moduli spaces of Gieseker-semistable torsion sheaves on the product of two projective lines, revealing birational transformations.

## Contribution

It connects wall-crossing in Bridgeland stability to known birational transformations of Simpson moduli spaces on b^1 b^1, providing new insights into their structure.

## Key findings

- Identifies specific wall-crossing maps corresponding to birational transformations.
- Recovers known birational transformations via Bridgeland wall-crossing.
- Analyzes stability conditions on b^1 b^1 and their impact on moduli spaces.

## Abstract

We investigate the wall-crossing behavior as Bridgeland moduli spaces for some Simpson moduli spaces of Gieseker-semistable torsion sheaves on $\mathbb{P}^1\times \mathbb{P}^1$ with linear Hilbert polynomial. In particular, we recover some of the birational transformations of these spaces described by Maican and Chung-Moon as wall-crossing maps in certain slices of $Stab(\mathbb{P}^1\times \mathbb{P}^1)$.