Birational geometry of symplectic quotient singularities

• 成果类型: Article
• 署名作者: Bellamy, Gwyn; Craw, Alastair
• 署名单位: University of Glasgow; University of Bath
• 刊物名称: INVENTIONES MATHEMATICAE
• ISSN/ISSBN: 0020-9910
• DOI: 10.1007/s00222-020-00972-9
• 发表日期: 2020
• 页码: 399-468
• 关键词: marsden-weinstein reductions; quiver varieties; poisson deformations; algebraic-families; MODULI SPACES; ale spaces; REPRESENTATIONS; flops; RESOLUTIONS; map

摘要

For a finite subgroup Gamma subset of SL(2,C)Gamma \subset \mathrm {SL}(2,\mathbb {C})$$\end{document} and for n >= 1, we use variation of GIT quotient for Nakajima quiver varieties to study the birational geometry of the Hilbert scheme of n points on the minimal resolution S of the Kleinian singularity C2/Gamma. It is well known that X:=Hilb[n](S)is a projective, crepant resolution of the symplectic singularity C2n/Gamma n\ where Gamma n=Gamma wreath product Sn is the wreath product. We prove that every projective, crepant resolution of C2n/Gamma n can be realised as the fine moduli space of theta-stable pi-modules for a fixed dimension vector, where pi is the framed preprojective algebra of Gamma and theta is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of theta-stability conditions to birational transformations of X over C2n/Gamma n. As a corollary, we describe completely the ample and movable cones of X over C2n/Gamma n\, and show that the Mori chamber decomposition of the movable cone is determined by an extended Catalan hyperplane arrangement of the ADE root system associated to Gamma by the McKay correspondence. In the appendix, we show that morphisms of quiver varieties induced by variation of GIT quotient are semismall, generalising a result of Nakajima in the case where the quiver variety is smooth.