Observation Nuclei in Affine Geometry: Symbolic Powers, Fat Points, and Infinitesimal Recovery

[2509.25212] Observation Nuclei in Affine Geometry: Symbolic Powers, Fat Points, and Infinitesimal Recovery Skip to main content Search Submit Donate Log in Mathematics > Commutative Algebra arXiv:2509.25212 (math) [Submitted on 21 Sep 2025 ( v1 ), last revised 25 Aug 2026 (this version, v6)] Title: Observation Nuclei in Affine Geometry: Symbolic Powers, Fat Points, and Infinitesimal Recovery Authors: Dang Vo Phuc View PDF HTML (experimental) Abstract: This paper integrates point-dependent infinitesimal data into affine geometry through quantale nuclei and finitary weak ideal systems. For a subset $X \subseteq k^n$ and a bounded function $\nu:X \to \mathbb{N}_{>0}$, we study the observation nucleus $\Phi^*_{X,\nu}(A) = \bigcap_{a\in X} (A+\mathfrak{m}_a^{\nu(a)})$. This construction organizes several classical interfaces within one point-dependent operator: closed observations recover symbolic powers through the Zariski-Nagata theorem, and finite observations restrict on ideals to translation by a fat-point ideal through the Chinese remainder theorem. The principal uniform identity is the exact relative Nullstellensatz $\sqrt{\Phi^*_{X,\nu}(I)} = \mathcal{I}(X\cap V_{\mathrm{aff}}(I))$, which separates the reduced geometry determined by the observation set from the retained infinitesimal multiplicities determined by $\nu$. This identity yields a density criterion for fixed prime ideals and identifies their space with the sobrification of the observed point set. For constant full-affine observation order, the descending tower has a common radical $\sqrt{I}$, and the nilpotent Nullstellensatz of Eisenbud and Hochster implies finite-stage stabilization: for every ideal $I$, there exists $e \geq 1$ such that $\Phi^*_q(I) = I$ for all $q \geq e$. The observation formalism therefore gives a simultaneous closed-point congruence interpretation of this classical theorem. Finally, the full affine nucleus is not a fixed-ideal translation in any positive dimension, and an explicit unbounded example shows that boundedness cannot be omitted from the general radical formula. Comments: 23 pages. In this new version, the results have been completely restructured compared to previous ones, and the title has been adjusted accordingly. Comments are always welcome! Subjects: Commutative Algebra (math.AC) ; Algebraic Geometry (math.AG); Algebraic Topology (math.AT); Rings and Algebras (math.RA) MSC classes: Primary 13A15, 14A15, Secondary 06F07, 13E05, 14A05, 14B05 Cite as: arXiv:2509.25212 [math.AC] (or arXiv:2509.25212v6 [math.AC] for this version) https://doi.org/10.48550/arXiv.2509.25212 Focus to learn more arXiv-issued DOI via DataCite Submission history From: Vo Phuc Dang [ view email ] [v1] Sun, 21 Sep 2025 23:21:52 UTC (17 KB) [v2] Wed, 15 Oct 2025 04:30:36 UTC (18 KB) [v3] Sat, 4 Apr 2026 14:58:26 UTC (19 KB) [v4] Tue, 21 Apr 2026 15:31:03 UTC (21 KB) [v5] Tue, 28 Apr 2026 04:09:44 UTC (21 KB) [v6] Tue, 25 Aug 2026 04:55:27 UTC (20 KB) Full-text links: Access Paper: View PDF HTML (experimental) TeX Source view license Current browse context: math.AC < prev | next > new | recent | 2025-09 Change to browse by: math math.AG math.AT math.RA 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. 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