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• W3023967513 abstract "Abstract For a finite subgroup $$Gamma subset mathrm {SL}(2,mathbb {C})$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>Γ</mml:mi> <mml:mo>⊂</mml:mo> <mml:mi>SL</mml:mi> <mml:mo>(</mml:mo> <mml:mn>2</mml:mn> <mml:mo>,</mml:mo> <mml:mi>C</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:math> and for $$nge 1$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>n</mml:mi> <mml:mo>≥</mml:mo> <mml:mn>1</mml:mn> </mml:mrow> </mml:math> , 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 $$mathbb {C}^2/Gamma $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mn>2</mml:mn> </mml:msup> <mml:mo>/</mml:mo> <mml:mi>Γ</mml:mi> </mml:mrow> </mml:math> . It is well known that $$X:={{,mathrm{{mathrm {Hilb}}},}}^{[n]}(S)$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:mi>X</mml:mi> <mml:mo>:</mml:mo> <mml:mo>=</mml:mo> <mml:msup> <mml:mrow> <mml:mrow> <mml:mspace /> <mml:mi>Hilb</mml:mi> <mml:mspace /> </mml:mrow> </mml:mrow> <mml:mrow> <mml:mo>[</mml:mo> <mml:mi>n</mml:mi> <mml:mo>]</mml:mo> </mml:mrow> </mml:msup> <mml:mrow> <mml:mo>(</mml:mo> <mml:mi>S</mml:mi> <mml:mo>)</mml:mo> </mml:mrow> </mml:mrow> </mml:math> is a projective, crepant resolution of the symplectic singularity $$mathbb {C}^{2n}/Gamma _n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> , where $$Gamma _n=Gamma wr mathfrak {S}_n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> <mml:mo>=</mml:mo> <mml:mi>Γ</mml:mi> <mml:mo>≀</mml:mo> <mml:msub> <mml:mi>S</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> is the wreath product. We prove that every projective, crepant resolution of $$mathbb {C}^{2n}/Gamma _n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> can be realised as the fine moduli space of $$theta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>θ</mml:mi> </mml:math> -stable $$Pi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Π</mml:mi> </mml:math> -modules for a fixed dimension vector, where $$Pi $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Π</mml:mi> </mml:math> is the framed preprojective algebra of $$Gamma $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Γ</mml:mi> </mml:math> and $$theta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>θ</mml:mi> </mml:math> is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of $$theta $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>θ</mml:mi> </mml:math> -stability conditions to birational transformations of X over $$mathbb {C}^{2n}/Gamma _n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> . As a corollary, we describe completely the ample and movable cones of X over $$mathbb {C}^{2n}/Gamma _n$$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mrow> <mml:msup> <mml:mrow> <mml:mi>C</mml:mi> </mml:mrow> <mml:mrow> <mml:mn>2</mml:mn> <mml:mi>n</mml:mi> </mml:mrow> </mml:msup> <mml:mo>/</mml:mo> <mml:msub> <mml:mi>Γ</mml:mi> <mml:mi>n</mml:mi> </mml:msub> </mml:mrow> </mml:math> , 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 $$Gamma $$ <mml:math xmlns:mml=http://www.w3.org/1998/Math/MathML> <mml:mi>Γ</mml:mi> </mml:math> 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." @default.
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• W3023967513 title "Birational geometry of symplectic quotient singularities" @default.
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