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Variations of GIT quotients and dimer combinatorics for toric compound Du Val singularities

Yusuke Nakajima

TL;DR

The work develops a combinatorial framework linking dimer models and zigzag paths to the GIT stability parameter space Θ(Q)_ℝ for toric cDV singularities, enabling explicit tracking of stable representations through wall-crossings. It shows that chambers of Θ(Q)_ℝ correspond to sequences of zigzag paths, with walls determined by regions bounded by these paths and wall type reflecting slope relations, yielding flops or divisor-to-curve contractions under crossing. By constructing dimer models Γ_{a,b} for Δ(a,b) and analyzing the associated quivers, the authors describe how triangulations Δ_ω of the toric diagram encode the projective crepant resolutions 𝓜_ω and how stable perfect matchings implement these changes. The cD_4 case is treated with a parallel, albeit more intricate, analysis, producing a flop graph and GIT-region decomposition, and showing that all projective crepant resolutions are connected by wall-induced flops and derived-equivalent across walls. Overall, the paper provides a concrete, combinatorial mechanism to study moduli of stable representations and their geometric realizations as crepant resolutions, with explicit constructions and wall-crossing rules applicable to both cA-type and cD_4-type toric cDV singularities.

Abstract

A dimer model is a bipartite graph described on the real two-torus, and it gives the quiver as the dual graph. It is known that for any three-dimensional Gorenstein toric singularity, there exists a dimer model such that a GIT quotient parametrizing stable representations of the associated quiver is a projective crepant resolution of this singularity for some stability parameter. It is also known that the space of stability parameters has the wall-and-chamber structure, and for any projective crepant resolution of a three-dimensional Gorenstein toric singularity can be realized as the GIT quotient associated to a stability parameter contained in some chamber. In this paper, we consider dimer models giving rise to projective crepant resolutions of a toric compound Du Val singularity. We show that sequences of zigzag paths, which are special paths on a dimer model, determine the wall-and-chamber structure of the space of stability parameters. Moreover, we can track the variations of stable representations under wall-crossing using the sequences of zigzag paths.

Variations of GIT quotients and dimer combinatorics for toric compound Du Val singularities

TL;DR

The work develops a combinatorial framework linking dimer models and zigzag paths to the GIT stability parameter space Θ(Q)_ℝ for toric cDV singularities, enabling explicit tracking of stable representations through wall-crossings. It shows that chambers of Θ(Q)_ℝ correspond to sequences of zigzag paths, with walls determined by regions bounded by these paths and wall type reflecting slope relations, yielding flops or divisor-to-curve contractions under crossing. By constructing dimer models Γ_{a,b} for Δ(a,b) and analyzing the associated quivers, the authors describe how triangulations Δ_ω of the toric diagram encode the projective crepant resolutions 𝓜_ω and how stable perfect matchings implement these changes. The cD_4 case is treated with a parallel, albeit more intricate, analysis, producing a flop graph and GIT-region decomposition, and showing that all projective crepant resolutions are connected by wall-induced flops and derived-equivalent across walls. Overall, the paper provides a concrete, combinatorial mechanism to study moduli of stable representations and their geometric realizations as crepant resolutions, with explicit constructions and wall-crossing rules applicable to both cA-type and cD_4-type toric cDV singularities.

Abstract

A dimer model is a bipartite graph described on the real two-torus, and it gives the quiver as the dual graph. It is known that for any three-dimensional Gorenstein toric singularity, there exists a dimer model such that a GIT quotient parametrizing stable representations of the associated quiver is a projective crepant resolution of this singularity for some stability parameter. It is also known that the space of stability parameters has the wall-and-chamber structure, and for any projective crepant resolution of a three-dimensional Gorenstein toric singularity can be realized as the GIT quotient associated to a stability parameter contained in some chamber. In this paper, we consider dimer models giving rise to projective crepant resolutions of a toric compound Du Val singularity. We show that sequences of zigzag paths, which are special paths on a dimer model, determine the wall-and-chamber structure of the space of stability parameters. Moreover, we can track the variations of stable representations under wall-crossing using the sequences of zigzag paths.
Paper Structure (24 sections, 33 theorems, 68 equations, 22 figures)

This paper contains 24 sections, 33 theorems, 68 equations, 22 figures.

Table of Contents

  1. Introduction
  2. Moduli descriptions of crepant resolutions of singularities
  3. Toric compound Du Val singularities
  4. Summary of results
  5. The structure of the paper
  6. Preliminaries on dimer models and associated toric rings
  7. Dimer models
  8. Toric rings associated to dimer models
  9. Preliminaries on moduli spaces of quiver representations
  10. Quivers associated to dimer models
  11. Stability parameters and crepant resolutions
  12. Perfect matchings corresponding to torus orbits
  13. Wall-and-chamber structures
  14. Observations of boundary perfect matchings
  15. Descriptions of boundary perfect matchings
  16. ...and 9 more sections

Key Result

Theorem 1.1

Let $\Delta(a,b)$ be the toric diagram of the toric cDV singularity $R_{a,b}\coloneqq\mathbb{C}[x,y,z,w]/(xy-z^aw^b)$. Let $\Gamma\coloneqq\Gamma_{a,b}$ be a consistent dimer model associated to $\Delta(a,b)$ and $Q$ be the quiver obtained as the dual of $\Gamma$. Let $n\coloneqq a+b$, and consider such that under this correspondence, if a chamber $C\subset\Theta(Q)_\mathbb{R}$ corresponds to a s

Figures (22)

  • Figure 1: Toric diagrams of toric cDV singularities
  • Figure 2: An example of a dimer model
  • Figure 3: Zigzag paths on the dimer model given in Figure \ref{['ex_dimer1_basic']}
  • Figure 4: The zigzag polygon $\Delta(3,2)$ of the dimer model given in Figure \ref{['ex_dimer1_basic']}
  • Figure 5: The quiver associated to the dimer model given in Figure \ref{['ex_dimer1_basic']}
  • ...and 17 more figures

Theorems & Definitions (66)

  • Theorem 1.1: see Theorems \ref{['thm_main_wall']}, \ref{['thm_adjacentchamber_reflection']}, and Corollary \ref{['cor_chamber_zigzag_correspondence']} for more details
  • Definition 2.1
  • Definition 2.2: see IU_consistent
  • Example 2.3
  • Theorem 2.4: see e.g., GulIU_special
  • Example 2.5
  • Example 2.6
  • Definition 3.1: see King_moduli
  • Theorem 3.2: see IU_moduli, IU_anycrepant
  • Definition 3.3
  • ...and 56 more