Rational Singularities and Uniform Symbolic Topologies

TL;DR

This paper investigates when certain algebraic rings satisfy the uniform symbolic topology property, proving it for two-dimensional rational singularities and providing explicit bounds in specific surface singularity cases.

Contribution

It establishes the USTP for two-dimensional rational singularities and offers explicit, effective multipliers for particular classes of surface singularities.

Findings

Rational singularities in dimension two satisfy USTP.

Explicit multipliers D are provided for toric and du Val singularities.

Effective bounds are derived using geometric and invariant-based methods.

Abstract

Take $(R, \mathfrak{m})$ any normal Noetherian domain, either local or $\mathbb{N}$-graded over a field. We study the question of when $R$ satisfies the uniform symbolic topology property (USTP) of Huneke, Katz, and Validashti: namely, that there exists an integer $D>0$ such that for all prime ideals $P \subseteq R$, the symbolic power $P^{(Da)} \subseteq P^a$ for all $a >0$. Reinterpreting results of Lipman, we deduce that when $R$ is a two-dimensional rational singularity, then it satisfies the USTP. Emphasizing the non-regular setting, we produce explicit, effective multipliers $D$, working in two classes of surface singularities in equal characteristic over an algebraically closed field, using: (1) the volume of a parallelogram in $\mathbb{R}^2$ when $R$ is the coordinate ring of a simplicial toric surface; or (2) known invariants of du Val isolated singularities in characteristic zero due to Lipman.