Moduli of McKay quiver representations II: Gröbner basis techniques
By Alastair Craw, Diane Maclagan, Rekha R. Thomas. 2006
52 citations recorded by the source corpus.
Abstract In this paper we introduce several computational techniques for the study of moduli spaces of McKay quiver representations, making use of Grobner bases and toric geometry. For a finite abelian group G ⊂ GL ( n , k ) , 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.
Subfields: Discrete Mathematics and Combinatorics, Geometry and Topology
View Paper: https://www.semanticscholar.org/paper/aa2b71d8c5dd254f10f3a9670f8d392d569d1a3f
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In Discrete Mathematics and Combinatorics
- T. Bridgeland
- M. Wemyss
- G. Moore
- Michael R Douglas
- Raf Bocklandt
- Mark Gross
- Balázs Szendrői
- Bernd S Siebert
In Geometry and Topology
Citations in the corpus
- Superpotentials and higher order derivations (2008, background): “sit is convenienttosuppresstensorproductsignsas muchas possible, so we writeρi,jρforρi⊗ρj⊗ρ. In this notation detV = ρ 1,...,n. Denote the set of irreducible representations by Irr(G). Corollary 4.1 ([1]). Let Gbe a finite abelian subgroup of GL(n,C). Then the McKay quiver is the directed graph with ”
- Dirichlet Branes and Mirror Symmetry (2009, background): “The link between the physics and mathematics literature is made transparent in the construction of the coherent component by Craw-Maclagan-Thomas [103].”