2018/11/25 by Bellamy, Gwyn, Craw, Alastair · 2 citations
#Algebraic Geometry (math.AG) #FOS: Mathematics #Representation Theory (math.RT) #Symplectic Geometry (math.SG)
paper · doi:10.48550/arxiv.1811.09979
For a finite subgroup Γ⊂ SL(2,ℂ) 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/Γ. It is well known that X:=Hilb[n](S) is a projective, crepant resolution of the symplectic singularity ℂ2n/Γn, where Γn=Γ\wr\mathfrakSn is the wreath product. We prove that every projective, crepant resolution of ℂ2n/Γn can be realised as the fine moduli space of θ-stable Π-modules for a fixed dimension vector, where Π is the framed preprojective algebra of Γ and θ is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of θ-stability conditions to birational transformations of X over ℂ2n/Γn. As a corollary, we describe completely the ample and movable cones of X over ℂ2n/Γn, 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 Γ 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.