F-purity of hypersurfaces

In this note, we study F-purity of pairs, and show (as is the case with log canonicity) that F-purity is preserved at the F-pure threshold. We also characterize when F-purity is equivalent to sharp F-purity, an alternate notion of purity for pairs introduced in [Sch08]. These results on purity at the threshold generalize results appearing in [Har06, Sch08], and were expected to hold by many experts in the field. We conclude by extending results in [BMS09] on the set of all F-pure thresholds to the most general setting.