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On F-pure thresholds and quasi-F-purity of hypersurfaces

Jack J Garzella, Vignesh Jagathese

TL;DR

This work investigates how two invariants measure failure of $F$-purity for hypersurfaces in positive characteristic: the $F$-pure threshold $\mathrm{fpt}(f)$ and the quasi-$F$-pure height. It proves that for quasi-homogeneous isolated singularities, if $A/(f)$ is quasi-$F$-pure but not $F$-pure and the characteristic is odd with $p>n-2$, then $\mathrm{fpt}(f) \ge 1 - \frac{1}{p}$, showing quasi-$F$-pure singularities are as close to $F$-pure as possible; this extends prior Calabi–Yau results. The method relies on a Fedder-type criterion for quasi-$F$-purity and the discreteness of $F$-pure thresholds, avoiding heavier cohomological machinery. The paper also provides a detailed classification of Fermat-type hypersurfaces with respect to $F$-purity and quasi-$F$-purity, yielding explicit congruence criteria and examples, such as a quasi-$F$-pure height-$2$ Fermat quartic threefold in characteristic $7$, and motivates questions about quasi-$F$-splitting in moduli of quartic hypersurfaces.

Abstract

We show that quasi-$F$-pure but not $F$-pure isolated quasi-homogeneous hypersurface singularities necessarily have $F$-pure threshold $1 - \frac{1}{p}$. This extends work of Bhatt and Singh beyond the Calabi-Yau case. We also classify the (quasi)-$F$-purity of Fermat hypersurfaces.

On F-pure thresholds and quasi-F-purity of hypersurfaces

TL;DR

This work investigates how two invariants measure failure of -purity for hypersurfaces in positive characteristic: the -pure threshold and the quasi--pure height. It proves that for quasi-homogeneous isolated singularities, if is quasi--pure but not -pure and the characteristic is odd with , then , showing quasi--pure singularities are as close to -pure as possible; this extends prior Calabi–Yau results. The method relies on a Fedder-type criterion for quasi--purity and the discreteness of -pure thresholds, avoiding heavier cohomological machinery. The paper also provides a detailed classification of Fermat-type hypersurfaces with respect to -purity and quasi--purity, yielding explicit congruence criteria and examples, such as a quasi--pure height- Fermat quartic threefold in characteristic , and motivates questions about quasi--splitting in moduli of quartic hypersurfaces.

Abstract

We show that quasi--pure but not -pure isolated quasi-homogeneous hypersurface singularities necessarily have -pure threshold . This extends work of Bhatt and Singh beyond the Calabi-Yau case. We also classify the (quasi)--purity of Fermat hypersurfaces.

Paper Structure

This paper contains 8 sections, 15 theorems, 29 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. F-Pure Thresholds
  4. Witt Vectors
  5. Quasi-F-purity
  6. Main Result
  7. Fermat Type Hypersurfaces
  8. Questions on Quartic Threefolds

Key Result

Theorem 2.1

Let $(A,\mathfrak{m})$ be a regular local ring and $I \subset A$ an ideal. Then $A/I$ is $F$-pure if and only if $(I^{[p]}:I) \not\subset \mathfrak{m}^{[p]}$.

Theorems & Definitions (26)

  • Theorem 2.1: Fedder's Criterion FeddersCrit
  • Theorem 2.2: MustataTakagiWatanabeFThresholdsAndBernsteinSato Theorem 3.3,3.4
  • Theorem 3.1: Fedder Type Criterion for Quasi-$F$-Purity kawakami2022fedder
  • Corollary 3.1.1: kawakami2022fedder, corollary 4.19
  • Theorem 3.2
  • proof
  • Corollary 3.2.1
  • Example 3.3
  • Lemma 4.1
  • proof
  • ...and 16 more