Moduli of McKay quiver representations II: Gröbner basis techniques

Alastair Craw, Diane Maclagan, Rekha R. Thomas

Research output: Contribution to journalArticlepeer-review

6 Citations (SciVal)


In this paper we introduce several computational techniques for the study of moduli spaces of McKay quiver representations, making use of Groebner bases and toric geometry. For a finite abelian group G in 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]. 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.
Original languageEnglish
Pages (from-to)514-535
JournalJournal of Algebra
Issue number2
Publication statusPublished - 15 Oct 2007


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