Syzygies of Polarized Abelian Surfaces: A Reider-Type Criterion

[2608.27921] Syzygies of Polarized Abelian Surfaces: A Reider-Type Criterion Skip to main content Search Submit Donate Log in Mathematics > Algebraic Geometry arXiv:2608.27921 (math) [Submitted on 28 Aug 2026 ( v1 ), last revised 16 Sep 2026 (this version, v2)] Title: Syzygies of Polarized Abelian Surfaces: A Reider-Type Criterion Authors: Chunyi Li , Lei Song , Xiao Wang View PDF HTML (experimental) Abstract: Let $(X,L)$ be a polarized complex abelian surface with $L^2=2d$. We establish a Reider-type criterion for Property $N_p$. If $d\geq7$, then $L$ satisfies Property $N_0$ if and only if there is no elliptic curve $E\subseteq X$ with $L\cdot E\leq2$, with one explicitly described exception. If $p\geq1$ and $d\geq(p+2)^2+1$, then $L$ satisfies Property $N_p$ if and only if there is no elliptic curve $E\subseteq X$ with $L\cdot E\leq p+2$. The numerical bounds on $d$ are optimal for $p=0,1$. These results improve upon earlier work of Küronya--Lozovanu, Ito, and Rojas. We also construct a polarized abelian surface whose basepoint-freeness threshold is irrational. Comments: Comments are welcome Subjects: Algebraic Geometry (math.AG) Cite as: arXiv:2608.27921 [math.AG] (or arXiv:2608.27921v2 [math.AG] for this version) https://doi.org/10.48550/arXiv.2608.27921 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Lei Song [ view email ] [v1] Fri, 28 Aug 2026 04:52:43 UTC (107 KB) [v2] Wed, 16 Sep 2026 12:35:17 UTC (108 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AG < prev | next > new | recent | 2026-08 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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