Moduli of McKay quiver representations II: Grobner basis techniques

Craw, A., Maclagan, D. and Thomas, R.R. (2007) Moduli of McKay quiver representations II: Grobner basis techniques. Journal of Algebra, 316(2), pp. 514-535. (doi: 10.1016/j.jalgebra.2007.02.014)

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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 Yθ 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 Yθ, and, in the case where Yθ 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.

Item Type:Articles
Status:Published
Refereed:Yes
Glasgow Author(s) Enlighten ID:Craw, Dr Alastair
Authors: Craw, A., Maclagan, D., and Thomas, R.R.
College/School:College of Science and Engineering > School of Mathematics and Statistics > Mathematics
Journal Name:Journal of Algebra
ISSN:0021-8693
ISSN (Online):1090-266X
Published Online:06 March 2007
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