Moduli of McKay quiver representations II: Gröbner basis techniques

Abstract

In this paper we introduce several computational techniques for the study of moduli spaces of McKay quiver representations, making use of Gröbner bases and toric geometry. For a finite abelian group G ⊂ GL ( n , k ) , let Y θ be the coherent component of the moduli space of θ-stable representations of the McKay quiver. Our two main results are as follows: we provide a simple description of the quiver representations corresponding to the torus orbits of Y θ , and, in the case where Y θ equals Nakamura's G-Hilbert scheme, we present explicit equations for a cover by local coordinate charts. The latter theorem corrects the first result from Nakamura [I. Nakamura, Hilbert schemes of abelian group orbits, J. Algebraic Geom. 10 (4) (2001) 757–779]. The techniques introduced here allow experimentation in this subject and give concrete algorithmic tools to tackle further open questions. To illustrate this point, we present an example of a nonnormal G-Hilbert scheme, thereby answering a question raised by Nakamura.