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. We prove, under a boundedness assumption, which we show to hold on threefolds or for rank two sheaves on base manifolds of arbitrary dimension, 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 $\omega \in N^1(X{)}_{\mathbb{R}}$ 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.

51 pages; v2: added discussion concerning characteristic of the base field, fixed some typos and inconsistencies; removed "I" from title as second part has a new title; to appear in Geometry & Topology

References in corpus (1)

• A functorial construction of moduli of sheaves

Cited by in corpus (5)

• Bounded sets of sheaves on relative analytic spaces
• A Master Space for Moduli Spaces of Gieseker-Stable Sheaves
• Semi-continuity of Stability for Sheaves and Variation of Gieseker Moduli Spaces
• Moduli spaces of sheaves that are semistable with respect to a Kähler polarisation
• Moduli of parabolic sheaves and filtered Kronecker modules