Moduli of McKay quiver representations II: Gröbner basis techniques

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• 作者：Alastair Craw ; Diane Maclagan ; Rekha R. Thomas
• 关键词：McKay quiver ; Grö ; bner bases ; G-Hilbert scheme
• 刊名：Journal of Algebra
• 出版年：2007
• 出版时间：15 October 2007
• 年：2007
• 卷：316
• 期：2
• 页码：514-535
• 全文大小：270 K

文摘

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 , 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.