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Fedder type criteria for quasi-$F$-splitting II

Tatsuro Kawakami, Teppei Takamatsu, Shou Yoshikawa

TL;DR

The paper develops Fedder-type criteria for quasi-$F$-splitting to enable explicit computation of quasi-$F$-split heights across key classes of varieties, including Calabi–Yau hypersurfaces, bielliptic surfaces, Fano varieties, and rational double points. It applies these tools to derive structural results on inversion of adjunction, fiber products, and genus-one fibrations, and constructs Calabi–Yau hypersurfaces of arbitrary height in any characteristic by height-stratified families. By extending the Fedder framework to weighted multigraded rings and section rings, the authors unify methods for complete intersections in weighted projective spaces and provide concrete height formulas and examples (e.g., Fermat-type cases and wild conic bundles). The findings deliver both positive results (finite heights in many cases) and striking counterexamples (inversion of adjunction failures), enriching the understanding of positivity, singularities, and Frobenius-splitting phenomena in positive characteristic.

Abstract

In this paper, we apply Fedder-type criteria for quasi-$F$-splitting to provide explicit computations of quasi-$F$-split heights for Calabi-Yau hypersurfaces, bielliptic surfaces, Fano varieties, and rational double points. We also find interesting phenomena concerned with inversion of adjunction, fiber products, Fano varieties, and general fibers of fibrations.

Fedder type criteria for quasi-$F$-splitting II

TL;DR

The paper develops Fedder-type criteria for quasi--splitting to enable explicit computation of quasi--split heights across key classes of varieties, including Calabi–Yau hypersurfaces, bielliptic surfaces, Fano varieties, and rational double points. It applies these tools to derive structural results on inversion of adjunction, fiber products, and genus-one fibrations, and constructs Calabi–Yau hypersurfaces of arbitrary height in any characteristic by height-stratified families. By extending the Fedder framework to weighted multigraded rings and section rings, the authors unify methods for complete intersections in weighted projective spaces and provide concrete height formulas and examples (e.g., Fermat-type cases and wild conic bundles). The findings deliver both positive results (finite heights in many cases) and striking counterexamples (inversion of adjunction failures), enriching the understanding of positivity, singularities, and Frobenius-splitting phenomena in positive characteristic.

Abstract

In this paper, we apply Fedder-type criteria for quasi--splitting to provide explicit computations of quasi--split heights for Calabi-Yau hypersurfaces, bielliptic surfaces, Fano varieties, and rational double points. We also find interesting phenomena concerned with inversion of adjunction, fiber products, Fano varieties, and general fibers of fibrations.

Paper Structure

This paper contains 19 sections, 36 theorems, 204 equations, 1 table.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Notation and terminologies
  4. Quasi-F-splitting
  5. Fedder type criterion for quasi-$F$-splitting
  6. Quasi-$F$-splitting for weighted multigraded rings
  7. Quasi-F-split height for weighted multigraded rings
  8. Fedder type criteria for projective varieties
  9. Applications of Fedder type criteria
  10. Inversion of adjunction
  11. Fiber products
  12. Smoothness of genus one fibrations
  13. Existence of Calabi--Yau varieties of arbitrary height in any characteristic
  14. Proof.
  15. Proof.
  16. ...and 4 more sections

Key Result

Lemma 2.1

Let $f \colon Y \to X$ be an étale morphism of $F$-finite ${\mathbb F}_p$-schemes. Then we have

Theorems & Definitions (92)

  • Lemma 2.1
  • proof
  • Proposition 2.2
  • proof
  • Definition 2.3: Yobuko2
  • Remark 2.4
  • Proposition 2.5
  • proof
  • Remark 2.6
  • Proposition 2.7
  • ...and 82 more