Moduli of McKay quiver representations II: Grobner basis techniques

In this paper we introduce several computational techniques for the study of moduli spaces of McKay quiver representations, making use of Grobner bases and toric geometry. For a finite abelian group G C GL(n, k), let Y-theta be the coherent component of the moduli space of theta-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-theta, and, in the case where Y-theta 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. (c) 2007 Elsevier Inc. All rights reserved.