Moduli of McKay quiver representations I: the coherent component

Abstract

For a finite abelian group G ⊂ GL (n, k ), we describe the coherent component Yθ of the moduli space ℳθ of θ-stable McKay quiver representations. This is a not-necessarily-normal toric variety that admits a projective birational morphism Y θ → A k n / G obtained by variation of Geometric Invariant Theory quotient. As a special case, this gives a new construction of Nakamura's G-Hilbert scheme HilbG that avoids the (typically highly singular) Hilbert scheme of |G|-points in A k n . To conclude, we describe the toric fan of Yθ and hence calculate the quiver representation corresponding to any point ofYθ.