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

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.