We introduce a notion of stability for sheaves with respect to several polarisations that generalises the usual notion of Gieseker stability. Under a boundedness assumption which we show to hold on threefolds or for rank two sheaves on base manifolds of arbitrary dimension, we prove that semistable sheaves have a projective coarse moduli space that depends on a natural stability parameter. We then give two applications of this machinery. First, we show that given a real ample class on a smooth projective threefold there exists a projective moduli space of sheaves that are Gieseker semistable with respect to . Second, we prove that given any two ample line bundles on the corresponding Gieseker moduli spaces are related by Thaddeus flips.
"Variation of Gieseker moduli spaces via quiver GIT." Geom. Topol. 20 (3) 1539 - 1610, 2016. https://doi.org/10.2140/gt.2016.20.1539