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

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 $Stab(\mathbb{P}^1\times \mathbb{P}^1)$.