Zariski-Nagata Theorems for Singularities and the Uniform Izumi-Rees Property

TL;DR

This paper introduces the Uniform Izumi-Rees Property in Noetherian rings and applies it to establish containment relationships among symbolic powers of ideals in normal domains, advancing multiplicity theory and singularity analysis.

Contribution

It defines the Uniform Izumi-Rees Property and proves its implications for symbolic power containments in normal domains over fields.

Findings

Existence of a constant C for symbolic power containments in normal domains.

Establishment of new relationships between symbolic powers and multiplicity theory.

Application of the property to singularity analysis in algebraic geometry.

Abstract

We introduce and explore the Uniform Izumi-Rees Property in Noetherian rings with applications to multiplicity theory and containment relationships among symbolic powers of ideals. As an application, we prove that if $R$ is a normal domain essentially of finite type over a field, there exists a constant $C$ so that for all prime ideals $\mathfrak{p}\subseteq \mathfrak{q}\in\mbox{Spec}(R)$, if $\mathfrak{p}\subseteq \mathfrak{q}^{(t)}$, then for all $n\in\mathbb{N}$, there is a containment of symbolic powers $\mathfrak{p}^{(Cn)}\subseteq \mathfrak{q}^{(tn)}$.