Birational geometry of symplectic quotient singularities
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Birational geometry of symplectic quotient singularities Gwyn Bellamy1 · Alastair Craw2
Received: 4 January 2019 / Accepted: 9 April 2020 © The Author(s) 2020
Abstract For a finite subgroup ⊂ 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 / . It is well known that X := Hilb[n] (S) is a projective, crepant resolution of the symplectic singularity C2n / n , where n = Sn is the wreath product. We prove that every projective, crepant resolution of C2n / n can be realised as the fine moduli space of θ -stable -modules for a fixed dimension vector, where is the framed preprojective algebra of 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 / n . As a corollary, we describe completely the ample and movable cones of X over C2n / 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 by the McKay correspondence. In the appendix, we show that morphisms
B Alastair Craw
[email protected] http://people.bath.ac.uk/ac886/ Gwyn Bellamy [email protected] http://www.maths.gla.ac.uk/~gbellamy/
1
School of Mathematics and Statistics, University of Glasgow, University Place, Glasgow G12 8QQ, UK
2
Department of Mathematical Sciences, University of Bath, Claverton Down, Bath BA2 7AY, UK
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G. Bellamy, A. Craw
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. Contents 1 Introduction . . . . . . . . . . . . . . . . . . . . 2 A wall-and-chamber structure . . . . . . . . . . . 3 Étale local normal form for quiver varieties . . . . 4 Quiver varieties for the framed McKay quiver . . 5 Variation of GIT quotient in the cone F . . . . . . 6 Linearisation map to the movable cone . . . . . . 7 Reflection functors and the Namikawa Weyl group Appendix A. Variation of GIT for quiver varieties . . References . . . . . . . . . . . . . . . . . . . . . . .
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1 Introduction For a finite subgroup ⊂ SL(2, C), let S → C2 / denote the minimal resolution of the corresponding Kleinian singularity. The well-known paper by Kronheimer [44] realises S as a hyperkähler quotient, describes the ample cone of S as a Weyl chamber of the root system of type ADE associated to by the McKay correspondence, an
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