Variation of Gieseker moduli spaces via quiver GIT

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 ω∈N1(X)R on a smooth projective threefold X there exists a projective moduli space of sheaves that are Gieseker semistable with respect to ω. Second, we prove that given any two ample line bundles on X the corresponding Gieseker moduli spaces are related by Thaddeus flips..