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arXiv · 1605.06642

A Master Space for Moduli Spaces of Gieseker-Stable Sheaves

Daniel Greb·Julius Ross·Matei Toma
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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 $σ\in \mathbb{Q}_{\geq 0}^N\setminus\{0\}$. The set of semi stable sheaves admits a projective moduli space $\mathcal M_σ$. We prove that given a finite collection of parameters $σ$, 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.

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Daniel Greb, Julius Ross, Matei Toma. 2016-05-21. A Master Space for Moduli Spaces of Gieseker-Stable Sheaves. https://doi.org/10.1007/s00031-018-9477-6

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