TY - JOUR

T1 - Moduli of McKay quiver representations II

T2 - Gröbner basis techniques

AU - Craw, Alastair

AU - Maclagan, Diane

AU - R. Thomas, Rekha

PY - 2007/10/15

Y1 - 2007/10/15

N2 - 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.

AB - 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.

UR - http://dx.doi.org/10.1016/j.jalgebra.2007.02.014,

U2 - 10.1016/j.jalgebra.2007.02.014,

DO - 10.1016/j.jalgebra.2007.02.014,

M3 - Article

VL - 316

SP - 514

EP - 535

JO - Journal of Algebra

JF - Journal of Algebra

SN - 0021-8693

IS - 2

ER -