Moduli of McKay quiver representations II: Gröbner basis techniques
Moduli of McKay quiver representations II: Gröbner basis techniques
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McKay 箭袋表示的模数 II：Gröbner 基础技术
DOI：
10.1016/j.jalgebra.2007.02.014
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发表时间：
2006
影响因子：
0.9
通讯作者：
Rekha R. Thomas
中科院分区：
文献类型：
--
作者：
Alastair Craw;D. Maclagan;Rekha R. Thomas
本文介绍了利用Gröbner基和环面几何研究McKay箭图表示的模空间的几种计算方法。对于有限交换群G⊂GL(n，k)，设Yθ是Mckay箭图的θ稳定表示的模空间的凝聚分支.我们的两个主要结果如下：我们给出了对应于Yθ环面轨道的箭图表示的简单描述，并且在Yθ等于Nakamura的G-Hilbert格式的情况下，我们给出了局部坐标图覆盖的显式方程。后一个定理修正了Nakamura[I.Nakamura，交换群轨道的Hilbert格式，J.Algebraic Geom]的第一个结果。10(4)(2001)757-779]。这里介绍的技术允许在这个主题中进行实验，并提供具体的算法工具来解决进一步的开放问题。为了说明这一点，我们给出了一个非正规G-Hilbert格式的例子，从而回答了Nakamura提出的一个问题。
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 Yθbe 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 Yθequals 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.