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

TL;DR

The paper develops a uniform framework to compare symbolic powers, ordinary powers, and multiplicities in normal domains essentially of finite type over a field. By leveraging Rees valuations, Gaussian extensions, equimultiplicity theory, and homogenization, it proves the Uniform Izumi–Rees Property and derives uniform containments that extend Zariski–Nagata-type results to singular settings. It establishes explicit bounds via projective closures and base-field reductions, yielding a uniform constant controlling $e(R_{\fq}/fR_{\fq})$ against $\ord_{\fq}(f)$ and translating to symbolic-power containments $\fp^{(Cn)}\subseteq\fq^{(tn)}$ for suitable $C,t$. The results connect multiplicity theory, valuation theory, and uniform powers/topology, with broad implications for containment relations among symbolic and ordinary powers, and for singularity theory in algebraic geometry. The work also provides a pathway to transfer the algebraically closed-field results to arbitrary base fields through separable and purely inseparable extensions and normalization techniques.

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 $\mathfrak{p}^{(Cn)}\subseteq \mathfrak{q}^{(tn)}$.

Theorems & Definitions (58)

• theorem 1
• remark 1
• theorem 2: Corollary of the Uniform Symbolic Topology Theorem, ELSHHComparisonMaSchwedeSymbolicMuryamaSymbolic
• definition 1
• theorem 3: HunekeUniformBounds
• remark 2
• remark 3
• theorem 4: Properties Rees Valuations
• lemma 1
• proof