A Master Space for Moduli Spaces of Gieseker-Stable Sheaves

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 $\mathcal M_{\sigma}$\mathrm{.}Weprovethatgivenafinitecollectionofparameters$\sigma$, 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.