F-thresholds of hypersurfaces

TL;DR

This paper investigates the properties of F-thresholds of hypersurfaces in regular rings, proving their discreteness and rationality, and explores limits of F-pure thresholds, drawing parallels to conjectures in algebraic geometry.

Contribution

It establishes the discreteness and rationality of F-thresholds for hypersurfaces in F-finite regular rings and analyzes limits of F-pure thresholds, connecting to conjectures in the field.

Findings

F-thresholds of hypersurfaces are discrete and rational.

Limits of F-pure thresholds of principal ideals are F-pure thresholds and rational.

Analogues of Shokurov and Kollár conjectures are proposed for F-pure thresholds.

Abstract

We continue our study of F-thresholds begun in math/0607660 by an in depth analysis of the hypersurface case. We use the D--module theoretic description of generalized test ideals which allows us to show that in any F--finite regular ring the F-thresholds of hypersurfaces are discrete and rational (in math/0607660 the finite type over a field case was shown for arbitrary ideals). Furthermore we show that any limit of F-pure thresholds of principal ideals in bouneded dimension is again an F-pure-threshold, hence in particular the limit is rational. The study of the set of F-pure-thresholds leads to natural analogs of conjectures of Shokurov and Koll\'{a}r (for log canonical thresholds) in the case of F-pure-thresholds.