On the Equality of Ordinary and Symbolic Powers of Ideals

TL;DR

This paper investigates conditions under which ordinary and symbolic powers of prime ideals in regular local rings are equal, providing evidence for specific classes like monomial curves and low multiplicity rings.

Contribution

It offers new insights and partial evidence supporting the equality of ordinary and symbolic powers for certain classes of prime ideals.

Findings

Equality holds for monomial curve ideals.

Equality holds for rings with low multiplicities.

Supports the conjecture in specific cases.

Abstract

We consider the following question concerning the equality of ordinary and symbolic powers of ideals. In a regular local ring, if the ordinary and symbolic powers of a one-dimensional prime ideal are the same up to its height, then are they the same for all powers? We provide supporting evidence of a positive answer for classes of prime ideals defining monomial curves or rings of low multiplicities.