Abstract
In the setting of a variety X admitting a tilting bundle T we consider the problem of constructing X as a quiver GIT quotient of the algebra A:=mathrm{End}_{X}(T)^{mathrm{op}}. We prove that if the tilting equivalence restricts to a bijection between the skyscraper sheaves of X and the closed points of a quiver representation moduli functor for A=mathrm{End}_{X}(T)^{mathrm{op}} then X is indeed a fine moduli space for this moduli functor, and we prove this result without any assumptions on the singularities of X. As an application we consider varieties which are projective over an affine base such that the fibres are of dimension 1, and the derived pushforward of the structure sheaf on X is the structure sheaf on the base. In this situation there is a particular tilting bundle on X constructed by Van den Bergh, and our result allows us to reconstruct X as a quiver GIT quotient for an easy to describe stability condition and dimension vector. This result applies to flips and flops in the minimal model program, and in the situation of flops shows that both a variety and its flop appear as moduli spaces for algebras produced from different tilting bundles on the variety. We also give an application to rational surface singularities, showing that their minimal resolutions can always be constructed as quiver GIT quotients for specific dimension vectors and stability conditions. This gives a construction of minimal resolutions as moduli spaces for all rational surface singularities, generalising the G-Hilbert scheme moduli space construction which exists only for quotient singularities.
Highlights
In this paper we will determine conditions for X to be a fine moduli space for the quiver representation moduli functor FA, (Section 2.6), and this will allow us to prove that X is a quiver GIT quotient for a specific stability condition and dimension vector in a large class of examples
Restrict to a bijection between FB(C) and FA(C) FB is naturally isomorphic to FA. This leads us to the idea of working with a moduli functor for which X is a fine moduli space instead of working with X itself, and we prove the following variant of Sekiya and Yamaura’s result
We state our main result, which follows from the previous lemmas, and we show that the moduli functor FX is represented by X
Summary
In this paper we will determine conditions for X to be a fine moduli space for the quiver representation moduli functor FA, (Section 2.6), and this will allow us to prove that X is a quiver GIT quotient for a specific stability condition and dimension vector in a large class of examples These examples include applications to the minimal model program and to resolutions of rational surface singularities. In the setting of smooth, projective 3-folds flops were constructed as components of moduli spaces and shown to be derived equivalent in the work of Bridgeland [4], and this work was extended to include projective 3-folds with Gorenstein terminal singularities by Chen [9] These results were reinterpreted more generally via tilting bundles by Van den Bergh [31]. He would like to thank an anonymous referee who pointed out an error in an earlier version of the paper, the correction of which led to the discussion in Appendix A
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