Variation of Gieseker moduli spaces via quiver GIT

Abstract

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 $\omega \in N^1{\left(X\right)}_{ℝ}$ on a smooth projective threefold $X$ there exists a projective moduli space of sheaves that are Gieseker semistable with respect to $\omega$. Second, we prove that given any two ample line bundles on $X$ the corresponding Gieseker moduli spaces are related by Thaddeus flips.