Involution on the Graded Grothendieck Ring of Varieties and $\mathbb{D}$-Singularities

[2508.17587] Involution on the Graded Grothendieck Ring of Varieties and $\mathbb{D}$-Singularities Skip to main content Search Submit Donate Log in Mathematics > Algebraic Geometry arXiv:2508.17587 (math) [Submitted on 25 Aug 2025 ( v1 ), last revised 26 Aug 2026 (this version, v2)] Title: Involution on the Graded Grothendieck Ring of Varieties and $\mathbb{D}$-Singularities Authors: Andrew Burke View PDF HTML (experimental) Abstract: We realize a graded variant $K_0(Var_k^{dim})$ of the Grothendieck ring of varieties as a quadratic extension of the subring $K_0(Var_k^{sp})$ spanned by classes of smooth and proper varieties. As such, there exists a natural involution $\mathbb{D}$ on $K_0(Var_k^{dim})$. We show that $\mathbb{D}$ commutes with the symmetric power operations $Sym^m$ up to zero divisors. Moreover, we study varieties which are smooth up to cut-and-paste relations, which we call $\mathbb{D}$-singular varieties, and we give applications to compactifications of varieties and the irrationality of Kapranov zeta functions. Comments: v2: minor changes, the final version that appeared in IMRN Subjects: Algebraic Geometry (math.AG) MSC classes: 14B05, 14G10 Cite as: arXiv:2508.17587 [math.AG] (or arXiv:2508.17587v2 [math.AG] for this version) https://doi.org/10.48550/arXiv.2508.17587 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Andrew Burke [ view email ] [v1] Mon, 25 Aug 2025 01:16:50 UTC (33 KB) [v2] Wed, 26 Aug 2026 18:33:06 UTC (33 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AG < prev | next > new | recent | 2025-08 Change to browse by: math References & Citations SciX 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)

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