$F$-invariants of diagonal hypersurfaces

Abstract:

In this note, we derive formulas for the $F$-pure threshold, higher jumping numbers, and test ideals of diagonal and Fermat hypersurfaces. For these hypersurfaces, we answer a question of Schwede regarding the denominators of $F$-pure thresholds, and obtain tight upper bounds for the number of higher jumping numbers. Our results are valid over all (or all but finitely many) characteristics, and therefore allow us to construct examples in which the characteristic $p$ setting is drastically different than that over $\mathbb {C}$.