$F$-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 C.

The $$F$$ F -pure threshold of a Calabi–Yau hypersurface

“…The deformation space in (4) refers to embedded deformations of X ⊂ P n ; if n 5, this coincides with the versal deformation space of X as an abstract variety (see Remark 4.7). The following example from [He2] shows that all possible values of h from (1) above are indeed attained:…”

“…for a proof, see[He2, Theorem 3.1].Proof of Theorem 4.1. If µ f (p) = p, then Lemma 3.4 (1) shows that µ f (q) = q for all q, and assertions (1-3) follow.…”

Local 𝔪-adic constancy of F-pure thresholds and test ideals

“…We point out that for large p or large bounds B, the algorithms given are not feasible. For more effective methods for computing F-pure thresholds in certain cases, we refer the reader to [10,11,12]. Additionally, we refer the reader to [19] for computations of test ideals in a more general (e.g., non-principal) setting.…”

Measuring Singularities with Frobenius: The Basics

“…Daniel Hernández has computed many examples of F -thresholds in his PhD thesis (c.f. [24]), including any "diagonal" hypersurfaces x a1 1 + · · · + x an n (see [26]). There is also an algorithm to compute the F -threshold of any binomial as well; see [27], [51].…”