Variation of Gieseker moduli spaces via quiver GIT

TL;DR

This paper generalizes Gieseker stability for sheaves using multiple polarisations, establishing projective moduli spaces that depend on a stability parameter, with applications to threefolds and relations between different Gieseker moduli spaces.

Contribution

It introduces a new stability notion for sheaves with multiple polarisations and proves the existence of associated projective moduli spaces under certain conditions.

Findings

Existence of projective moduli spaces for semistable sheaves with respect to multiple polarisations.

Construction of moduli spaces depending on a natural stability parameter.

Relation of Gieseker moduli spaces via Thaddeus-flips for different ample line bundles.

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.