# Symbolic powers of ideals defining F-pure and strongly F-regular rings

## TL;DR

This paper investigates the containment problem for symbolic and ordinary powers of radical ideals in regular rings, providing new results for F-pure and strongly F-regular quotient rings.

## Contribution

It proves Harbourne's conjecture for F-pure rings and establishes tighter containments for strongly F-regular rings, advancing understanding of symbolic power containments.

## Key findings

- Harbourne's conjecture holds when R/I is F-pure.
- Tighter containments are proven for strongly F-regular R/I.
- Provides uniform bounds for symbolic and ordinary power containments.

## Abstract

Given a radical ideal $I$ in a regular ring $R$, the Containment Problem of symbolic and ordinary powers of $I$ consists of determining when the containment $I^{(a)} \subseteq I^b$ holds. By work of Ein-Lazersfeld-Smith, Hochster-Huneke and Ma-Schwede, there is a uniform answer to this question, but the resulting containments are not necessarily best possible. We show that a conjecture of Harbourne holds when $R/I$ is F-pure, and prove tighter containments in the case when $R/I$ is strongly F-regular.