[Paper Review] Moduli of McKay quiver representations II: Groebner basis techniques

Alastair Craw, Diane Maclagan+1 more|arXiv (Cornell University)|Nov 27, 2006

Algebraic structures and combinatorial models11 references4 citations

TL;DR

This paper develops computational techniques using Gröbner bases and toric geometry to study moduli spaces of McKay quiver representations, particularly the coherent component $Y_\theta$ of the moduli space. It provides an algorithm to determine whether a $\theta$-stable $G$-constellation lies in $Y_\theta$, and when $Y_\theta$ is the $G$-Hilbert scheme, gives explicit local coordinate charts. The key contribution is constructing a nonnormal $G$-Hilbert scheme, answering a question of Nakamura about normality.

ABSTRACT

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_θbe 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_θequals 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.

Motivation & Objective

• To develop algorithmic tools for studying moduli spaces of McKay quiver representations using Gröbner bases and toric geometry.
• To characterize which $\theta$-stable $G$-constellations lie in the coherent component $Y_\theta$ of the moduli space.
• To provide explicit local coordinate charts for the $G$-Hilbert scheme $\operatorname{Hilb}^G$ when it coincides with $Y_\theta$.
• To resolve Nakamura's question about the normality of $\operatorname{Hilb}^G$ by constructing a counterexample.

Proposed method

• Use Gröbner theory to compute initial modules of the McKay module $M_G$ with respect to a weight vector $({\bf v}, {\bf w})$.
• Apply linear programming to find a minimizer $\mathbf{v} \in P^\vee_{\mathbf{w}}$ of the $\theta$-functional, ensuring $\theta \cdot \mathbf{v} \leq \theta \cdot \mathbf{v}'$ for all $\mathbf{v}' \in P^\vee_{\mathbf{w}}$.
• Construct the distinguished $\theta$-semistable $G$-constellation as the cyclic $A$-module $A / \operatorname{in}_{({\bf v},{\bf w})}(M_G)$.
• Define the initial ideal $J$ as the initial ideal of the lattice ideal $I_M$ with respect to a specific weight vector, such as $(22,10,16,50,31,21)$.
• Use Normaliz to test semigroup normality of the associated semigroup $A_J$ to determine whether $\operatorname{Hilb}^G$ is normal.
• Construct a counterexample using a subgroup $G \subset \operatorname{GL}(6,\Bbbk)$ isomorphic to $(\mathbb{Z}/5\mathbb{Z})^4$ with a nonnormal $G$-Hilbert scheme.

Experimental results

Research questions

• RQ1 Does the $G$-Hilbert scheme $\operatorname{Hilb}^G$ remain normal for all finite abelian subgroups $G \subset \operatorname{GL}(n,\Bbbk)$?
• RQ2 Can one algorithmically determine whether a $\theta$-stable $G$-constellation lies in the coherent component $Y_\theta$?
• RQ3 What are explicit local coordinate charts for the $G$-Hilbert scheme when it coincides with $Y_\theta$?
• RQ4 Are the moduli spaces $\mathcal{M}_\theta$ always toric varieties, as claimed by Sardo Infirri?
• RQ5 Can one construct a nonnormal $G$-Hilbert scheme using computational algebraic geometry techniques?

Key findings

• The paper constructs a nonnormal $G$-Hilbert scheme for $G \subset \operatorname{GL}(6,\Bbb{k})$ isomorphic to $(\mathbb{Z}/5\mathbb{Z})^4$, thereby answering Nakamura's question about normality in the negative.
• The semigroup $A_J$ associated to the ideal $J$ is not normal, as evidenced by the vector $(3,2,-3,1,-1,-2)$ lying in the rational cone but not in $A_J$.
• The ideal $J$ is the initial ideal of the lattice ideal $I_M$ with respect to the weight vector $(22,10,16,50,31,21)$, confirming it defines a $G$-cluster in $\operatorname{Hilb}^G$.
• The algorithm based on solving a linear program and computing an initial module provides a practical method to determine membership of $\theta$-stable $G$-constellations in $Y_\theta$.
• The result contradicts Sardo Infirri's claim that all $\mathcal{M}_\theta$ are toric varieties, as shown by counterexamples in Examples 4.12 and 5.7.
• The coherent component $Y_\theta$ is not always normal, as demonstrated by the nonnormality of $\operatorname{Hilb}^G$ in the constructed example.