Abstract
For a finite subgroup $$\Gamma \subset \mathrm {SL}(2,\mathbb {C})$$
Γ
⊂
SL
(
2
,
C
)
and for $$n\ge 1$$
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 $$\mathbb {C}^2/\Gamma $$
C
2
/
Γ
. It is well known that $$X:={{\,\mathrm{{\mathrm {Hilb}}}\,}}^{[n]}(S)$$
X
:
=
Hilb
[
n
]
(
S
)
is a projective, crepant resolution of the symplectic singularity $$\mathbb {C}^{2n}/\Gamma _n$$
C
2
n
/
Γ
n
, where $$\Gamma _n=\Gamma \wr \mathfrak {S}_n$$
Γ
n
=
Γ
≀
S
n
is the wreath product. We prove that every projective, crepant resolution of $$\mathbb {C}^{2n}/\Gamma _n$$
C
2
n
/
Γ
n
can be realised as the fine moduli space of $$\theta $$
θ
-stable $$\Pi $$
Π
-modules for a fixed dimension vector, where $$\Pi $$
Π
is the framed preprojective algebra of $$\Gamma $$
Γ
and $$\theta $$
θ
is a choice of generic stability condition. Our approach uses the linearisation map from GIT to relate wall crossing in the space of $$\theta $$
θ
-stability conditions to birational transformations of X over $$\mathbb {C}^{2n}/\Gamma _n$$
C
2
n
/
Γ
n
. As a corollary, we describe completely the ample and movable cones of X over $$\mathbb {C}^{2n}/\Gamma _n$$
C
2
n
/
Γ
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 $$\Gamma $$
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.
Birational geometry of del Pezzo fibrations with terminal quotient singularities
