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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.

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