## Illinois Journal of Mathematics

### Rational singularities and uniform symbolic topologies

Robert M. Walker

#### 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.

#### Article information

Source
Illinois J. Math., Volume 60, Number 2 (2016), 541-550.

Dates
Revised: 9 February 2017
First available in Project Euclid: 11 July 2017

https://projecteuclid.org/euclid.ijm/1499760021

Digital Object Identifier
doi:10.1215/ijm/1499760021

Mathematical Reviews number (MathSciNet)
MR3680547

Zentralblatt MATH identifier
1374.13032

#### Citation

Walker, Robert M. Rational singularities and uniform symbolic topologies. Illinois J. Math. 60 (2016), no. 2, 541--550. doi:10.1215/ijm/1499760021. https://projecteuclid.org/euclid.ijm/1499760021

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