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

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 ${\Phi }_{X,\nu }^{\ast }(A)={\bigcap }_{a\in X}(A+{\mathfrak{m}}_a^{\nu (a)})$. 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 $\sqrt{{\Phi }_{X,\nu }^{\ast }(I)}=\mathcal{I}(X\cap V_{\mathrm{aff}}(I))$, 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 $e\ge 1$ such that ${\Phi }_q^{\ast }(I)=I$ for all $q\ge e$. 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.

arXiv comment23 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!