On Modified Diagonal Cycles and the Beauville Decomposition of the Ceresa Cycle

[2510.07416] On Modified Diagonal Cycles and the Beauville Decomposition of the Ceresa Cycle Skip to main content Search Submit Donate Log in Mathematics > Algebraic Geometry arXiv:2510.07416 (math) [Submitted on 8 Oct 2025 ( v1 ), last revised 24 Aug 2026 (this version, v3)] Title: On Modified Diagonal Cycles and the Beauville Decomposition of the Ceresa Cycle Authors: Lucas Lagarde , Mohamed Moakher , Morena Porzio , James Rawson , Fernando Trejos Suárez View PDF HTML (experimental) Abstract: Let $C$ be a curve of genus $g \geq 2$, and let $J$ be its Jacobian. The choice of a degree 1 divisor $e$ on $C$ gives an embedding of $C$ into $J$; we denote by $[C]_{}^{e}\in \mathrm{CH}\left( J;\mathbb{Q} \right) $ the class in the Chow group of $J$ defined by its image. It is known that the vanishing of the Ceresa cycle $\mathrm{Cer}(C,e):=[C]^{e} - [-1]_* [C]^e$ is equivalent to both the vanishing of the 1st Beauville component $[C]_{(1)}^e$ and the vanishing of the 3rd Gross--Kudla--Schoen modified diagonal cycle $\Gamma^3(C,e) \in \mathrm{CH}(C^3;\mathbb{Q})$. We extend this result to show that the vanishing of the $s$-th Beauville component $[C]^e_{(s)}$ for $s \geq 1$ is equivalent to the vanishing of the $(s+2)$-nd modified diagonal cycle $\Gamma^{s + 2}(C, e) \in \mathrm{CH}(C^{s+2};\mathbb{Q})$. Moreover, we establish "successive vanishing" results for these cycles. We apply our results to study the rational (non)-triviality of $[C]^{e}_{(s)}$ in the special case $s = 2$. Finally in the $s=1$ case, we show an integral refinement to the original statement, relating the order of torsion of $\mathrm{Cer}(C,e) \in \mathrm{CH}(J;\mathbb{Z})$ to that of $\Gamma^3(C,e) \in \mathrm{CH}(C^3;\mathbb{Z})$. Comments: 34 pages, comments welcome. Final version, to appear in Épijournal de Géométrie Algébrique Subjects: Algebraic Geometry (math.AG) ; Number Theory (math.NT) MSC classes: 14C25, 14C15, 14H40 Cite as: arXiv:2510.07416 [math.AG] (or arXiv:2510.07416v3 [math.AG] for this version) https://doi.org/10.48550/arXiv.2510.07416 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Fernando Trejos Suarez [ view email ] [v1] Wed, 8 Oct 2025 18:15:32 UTC (40 KB) [v2] Tue, 11 Nov 2025 18:38:28 UTC (42 KB) [v3] Mon, 24 Aug 2026 18:00:12 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 | 2025-10 Change to browse by: math math.NT 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)

Related documents

• On Modified Diagonal Cycles and the Beauville Decomposition of the Ceresa Cycle#670396
• Two families of Hermite normal form simplices#1132019
• Spectrum of singularities, analytic torsion and Calabi--Yau degenerations#1070307
• Generalised Beauville Groups#407650
• On Hodge structures of compact complex manifolds with semistable degenerations#131225