ResearchResearchWe present a new relation between an invariant of singularities in characteristic zero (the log canonical threshold) and an invariant of singularities defined via the Frobenius morphism in positive characteristic (the F-pure threshold). Our methods rely on combining results of Hara and Yoshida with non-standard constructions.
This article describes an algorithm for computing the F-pure threshold of a binomial hypersurface; this algorithm remains valid in arbitrary characteristic, and generalizes earlier computations by Takafumi Shibuta and Shunsuke Takagi. The algorithm described in this article has recently been implemented in a Macaulay2 package by Sara Malec, Karl Schwede, and Emily Witt.
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 by Karl Schwede. We conclude by extending results describing the set of all F-pure thresholds to the most general setting.
This article considers the long-conjectured relationship between F-purity and log canonicity for polynomials over the complex numbers. We define a non-degeneracy condition under which log canonicity and dense F-pure type are equivalent, and we also show that log canonicity corresponds, after reduction to characteristic p, to F-purity for very general polynomials. Note: This result has recently been generalized by Shunsuke Takagi.
We describe the higher jumping numbers and generalized test ideals associated to diagonal hypersurfaces. Though these invariants are understood asymptotically (that is, as the characteristic tends to infinity), the results in this article allow one to produce examples in which the behavior in characteristic zero differs drastically from that in some fixed prime characteristic.
We give a description of the F-pure threshold associated to a homogeneous (under an arbitrary grading) polynomial with an isolated singularity at the origin. Our methods allow us to produce seemingly "minimal lists" for these invariants, and also allows us to answer a question of Karl Schwede and myself regarding certain arithmetic properties of these invariants when they differ from their expected values predicted by characteristic zero considerations.