A Master Space for Moduli Spaces of Gieseker-Stable Sheaves

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 $\sigma \in {\mathbb{Q}}_{\ge 0}^N\setminus \{0\}$. 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.

18 pages

References in corpus (3)

• A functorial construction of moduli of sheaves
• Variation of Gieseker moduli spaces via quiver GIT
• Semi-continuity of Stability for Sheaves and Variation of Gieseker Moduli Spaces

Cited by in corpus (1)

• Moduli of sheaves on $K3$'s and higher dimensional HK varieties