Symbolic Powers of Ideals Defining F-pure and Strongly F-regular Rings

Abstract

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.