Variations of GIT Quotients and Dimer Combinatorics for Toric Compound Du Val Singularities

[2309.16112] Variations of GIT Quotients and Dimer Combinatorics for Toric Compound Du Val Singularities Skip to main content Search Submit Donate Log in Mathematics > Algebraic Geometry arXiv:2309.16112 (math) [Submitted on 28 Sep 2023 ( v1 ), last revised 28 Aug 2026 (this version, v4)] Title: Variations of GIT Quotients and Dimer Combinatorics for Toric Compound Du Val Singularities Authors: Yusuke Nakajima View PDF HTML (experimental) Abstract: A dimer model is a bipartite graph described on the real two-torus, from which we can define the associated quiver. 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 a wall-and-chamber structure, and 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 variations of stable representations under wall-crossing using the sequences of zigzag paths. Subjects: Algebraic Geometry (math.AG) ; Combinatorics (math.CO); Representation Theory (math.RT) Cite as: arXiv:2309.16112 [math.AG] (or arXiv:2309.16112v4 [math.AG] for this version) https://doi.org/10.48550/arXiv.2309.16112 Focus to learn more arXiv-issued DOI via DataCite Journal reference: SIGMA 22 (2026), 083, 45 pages Related DOI : https://doi.org/10.3842/SIGMA.2026.083 Focus to learn more DOI(s) linking to related resources Submission history From: Yusuke Nakajima [ view email ] [via Journal Sigma as proxy] [v1] Thu, 28 Sep 2023 02:37:59 UTC (55 KB) [v2] Thu, 1 May 2025 03:17:47 UTC (56 KB) [v3] Wed, 26 Aug 2026 18:31:08 UTC (61 KB) [v4] Fri, 28 Aug 2026 05:06:11 UTC (61 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AG < prev | next > new | recent | 2023-09 Change to browse by: math math.CO math.RT References & Citations NASA ADS Google Scholar Semantic Scholar export BibTeX citation Bookmark Bibliographic Tools Bibliographic and Citation Tools Bibliographic Explorer Toggle Bibliographic Explorer ( What is the Explorer? ) Connected Papers Toggle Connected Papers ( What is Connected Papers? ) Litmaps Toggle Litmaps ( What is Litmaps? ) scite.ai Toggle scite Smart Citations ( What are Smart Citations? ) Code, Data, Media Code, Data and Media Associated with this Article alphaXiv Toggle alphaXiv ( What is alphaXiv? ) Links to Code Toggle CatalyzeX Code Finder for Papers ( What is CatalyzeX? ) DagsHub Toggle DagsHub ( What is DagsHub? ) GotitPub Toggle Gotit.pub ( What is GotitPub? ) Huggingface Toggle Hugging Face ( What is Huggingface? ) ScienceCast Toggle ScienceCast ( What is ScienceCast? ) Demos Demos Replicate Toggle Replicate ( What is Replicate? ) Spaces Toggle Hugging Face Spaces ( What is Spaces? ) Spaces Toggle TXYZ.AI ( What is TXYZ.AI? ) Related Papers Recommenders and Search Tools Link to Influence Flower Influence Flower ( What are Influence Flowers? ) Core recommender toggle CORE Recommender ( What is CORE? ) Author Venue Institution Topic About arXivLabs arXivLabs: experimental projects with community collaborators arXivLabs is a framework that allows collaborators to develop and share new arXiv features directly on our website. Both individuals and organizations that work with arXivLabs have embraced and accepted our values of openness, community, excellence, and user data privacy. arXiv is committed to these values and only works with partners that adhere to them. Have an idea for a project that will add value for arXiv's community? Learn more about arXivLabs . Which authors of this paper are endorsers? | Disable MathJax ( What is MathJax? ) We gratefully acknowledge support from our major funders , member institutions , and all contributors. About · Help · Contact · Subscribe · Copyright · Privacy · Accessibility · Operational Status (opens in new tab) Major funding support from

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