A Master Space for Moduli Spaces of Gieseker-Stable Sheaves

TL;DR

This paper introduces a master space framework for moduli spaces of Gieseker-stable sheaves, showing how different moduli spaces relate via GIT quotients and flips, extending known surface results to higher dimensions.

Contribution

It constructs a universal master space for multi-Gieseker stability, linking various moduli spaces through GIT and flips, and generalizes surface case results to arbitrary dimensions.

Findings

Existence of a master space Y for multiple stability parameters.

Moduli spaces are obtained as GIT quotients of Y.

Different Gieseker moduli spaces are connected via Thaddeus-flips.

Abstract

We consider a notion of stability for sheaves, which we call multi-Gieseker stability that depends on several ample polarisations $L_1, \dots, L_N$ and on an additional parameter $\sigma \in \mathbb{Q}_{\geq 0}^N\setminus\{0\}$. The set of semi stable sheaves admits a projective moduli space $\mathcal M_{\sigma}$. We prove that given a finite collection of parameters $\sigma$, there exists a sheaf- and representation-theoretically defined master space $Y$ such that each corresponding moduli space is obtained from $Y$ as a Geometric Invariant Theory (GIT) quotient. In particular, any two such spaces are related by a finite number of "Thaddeus-flips". As a corollary, we deduce that any two Gieseker-moduli space of sheaves (with respect to different polarisations $L_1$ and $L_2$) are related via a GIT-master space. This confirms an old expectation and generalises results from the surface case to arbitrary dimension.