Variational Geometric Invariant Theory and Moduli of Quiver Sheaves

Abstract

We are concerned with two applications of GIT. First, we prove that a geometric GIT quotient of an a ne variety X = Spec(A) by a reductive group G, where A is an almost factorial domain, is a Mori dream space, regardless of the codimension of the unstable locus. This includes an explicit description of the Picard number, the pseudoe ective cone, and the Mori chambers in terms of GIT. We apply the results to quiver moduli to show that they are Mori dream spaces if the quiver contains no oriented cycles, and if stability and semistability coincide. We give a formula for the Picard number in quiver terms. As a second application, we prove that geometric quotients of Mori dream spaces are Mori dream spaces as well, which again includes a description of the Picard number and the Mori chambers. Some examples are given to illustrate the results. The second instance where we use GIT, is the construction and variation of moduli spaces of quiver sheaves. To that end, we generalize the notion of multi{Gieseker semistability for coherent sheaves, introduced by Greb, Ross, and Toma, to quiver sheaves for a quiver Q. We construct coarse moduli spaces for semistable quiver sheaves using a functorial method that realizes these as subschemes of moduli spaces of representations of a twisted quiver, depending on Q, with relations. We also show the projectivity of the moduli space in the case when Q has no oriented cycles. Further, we construct moduli spaces of quiver sheaves which satisfy a given set of relations as closed subvarieties. Finally, we investigate the parameter dependence of the moduli.