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Moduli of McKay quiver representations. II: Gröbner basis techniques. (English) Zbl 1161.14038

Let \(G\) be a finite abelian group acting linearly on \(k^n\), where \(k\) is an algebraically closed field, and the order of \(G\) is invertible in \(k\) (so the action can be diagonalised). The present paper is a sequel to [Proc. Lond. Math. Soc. (3) 95, No. 1, 179–198 (2007; Zbl 1140.14046)], where the authors constructed expilitly an irreducible component \(Y_{\theta}\) (called the coherent component) of the moduli space of \(\theta\)-stable representations of the McKay quiver of \(G\).
Passing from a \(\theta\)-stable quiver representation to a \(G\)-constellation (i.e. a \(G\)-equivariant \(S\)-module isomorphic to the regular \(G\)-module \(kG\), where \(S\) is the coordinate ring of the \(G\)-module \(k^n\)), the authors can make use of Gröbner basis theory. They determine whether a given \(\theta\)-stable \(G\)-constellation corresponds to a point on the coherent component \(Y_{\theta}\). In the case when \(Y_{\theta}\) equals Nakamura’s \(G\)-Hilbert scheme, they present explicit equations for a cover by local coordinate charts. The computational techniques introduced here are applied to construct a subgroup of \(GL(6,k)\) for which the \(G\)-Hilbert scheme is not normal.

MSC:

14M25 Toric varieties, Newton polyhedra, Okounkov bodies
13P10 Gröbner bases; other bases for ideals and modules (e.g., Janet and border bases)
16G20 Representations of quivers and partially ordered sets

Citations:

Zbl 1140.14046
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References:

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