A canonical Fano threefold has Fano index $\leq 66$

[2508.16364] A canonical Fano threefold has Fano index $\leq 66$ Skip to main content Search Submit Donate Log in Mathematics > Algebraic Geometry arXiv:2508.16364 (math) [Submitted on 22 Aug 2025 ( v1 ), last revised 27 Aug 2026 (this version, v3)] Title: A canonical Fano threefold has Fano index $\leq 66$ Authors: Chen Jiang , Haidong Liu View PDF HTML (experimental) Abstract: We show that the $\mathbb{Q}$-Fano index of a canonical weak Fano $3$-fold is at most $66$. This upper bound is optimal and gives an affirmative answer to a conjecture of Chengxi Wang in dimension $3$. During the proof, we establish a new Riemmann--Roch formula for canonical $3$-folds and provide a detailed study of non-isolated singularities on canonical Fano $3$-folds, concerning both their local and global properties. Our proof also involves a Kawamata--Miyaoka type inequality and geometry of foliations of rank $2$ on canonical Fano $3$-folds. Comments: 50 pages, 5 tables. Comments are welcome! This preprint supersedes our old preprint arXiv:2505.19541 . Ver2: We clarify an inaccurate statement on crepant divisors in Ver1 pointed out by Professor Yuri Prokhorov and introduce the concept of non-split crepant divisors (Definition 2.3), main results are not affected. Ver3: The published version Subjects: Algebraic Geometry (math.AG) MSC classes: Primary 14J45, Secondary 14J30, 14J10, 14E30, 14C20, 14C40 Cite as: arXiv:2508.16364 [math.AG] (or arXiv:2508.16364v3 [math.AG] for this version) https://doi.org/10.48550/arXiv.2508.16364 Focus to learn more arXiv-issued DOI via DataCite Journal reference: Proc. Lond. Math. Soc. (3) 133 (2026), no. 2, Paper No. e70195 Related DOI : https://doi.org/10.1112/plms.70195 Focus to learn more DOI(s) linking to related resources Submission history From: Haidong Liu [ view email ] [v1] Fri, 22 Aug 2025 13:15:52 UTC (45 KB) [v2] Mon, 20 Oct 2025 14:27:59 UTC (46 KB) [v3] Thu, 27 Aug 2026 03:56:20 UTC (48 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) Major funding support from (opens in new tab) (opens in new tab) (opens in new tab)

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