Moduli of McKay quiver representations II: Groebner basis techniques

Details Moduli of McKay quiver representations II: Groebner basis techniques Moduli of McKay quiver representations II: Groebner basis techniques

2006-11-27

arxiv.org/abs/math/0611840v2

Journal of Algebra 316 (2007) 514-535

Mathematics

Algebraic Geometry

21 pages; explicit description of the universal family over Hilb^G added, typos corrected. To appear in J. Algebra

Scientific paper

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.

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