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

Tatsuro Kawakami, Teppei Takamatsu, Shou Yoshikawa

TL;DR

This work generalizes the classical Fedder criterion from $F$-splitting to quasi-$F$-splitting by introducing a Witt-vector framework that uses only $W_2$-data. The authors build a concrete, computable mechanism via $Q_{S,n}$ and the Delta maps to determine the quasi-$F$-split height ht$(X)$ (and, in Calabi–Yau cases, the Artin–Mazur height) through a single, explicit sequence of ideals $\{I_n\}$ with ht$(R/f)=\inf\{n\mid I_n\nsubseteq \mathfrak{m}^{[p]}\}$. They extend the criterion to graded complete intersections, enabling straightforward calculations in weighted/projective settings and deriving a Calabi–Yau height equality in this graded context. The paper then demonstrates significant applications, including the construction of Calabi–Yau varieties of arbitrarily high Artin–Mazur height over $\mathbb{F}_2$ and explicit defining equations for quartic K3 surfaces over $\mathbb{F}_3$ realizing a full range of heights, thereby highlighting both the theoretical depth and the algorithmic practicality of the approach. Overall, the results deepen the understanding of how Frobenius-related splitting properties govern formal-group heights and provide new tools to explore height phenomena in Calabi–Yau geometry in positive characteristic.

Abstract

Yobuko recently introduced the notion of quasi-$F$-splitting and quasi-$F$-split heights, which generalize and quantify the notion of Frobenius-splitting, and proved that quasi-$F$-split heights coincide with Artin-Mazur heights for Calabi-Yau varieties. In this paper, we prove Fedder type criteria for quasi-$F$-splittings of complete intersections, and in particular, obtain a simple formula to compute Artin-Mazur heights of Calabi-Yau hypersurfaces. As one of its applications, we prove that there exist Calabi-Yau varieties of arbitrarily high Artin-Mazur height over $\mathbb{F}_2$. We also give explicit defining equations of quartic K3 surfaces over $\mathbb{F}_{3}$ realizing all the possible Artin-Mazur heights.

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

TL;DR

This work generalizes the classical Fedder criterion from -splitting to quasi--splitting by introducing a Witt-vector framework that uses only -data. The authors build a concrete, computable mechanism via and the Delta maps to determine the quasi--split height ht (and, in Calabi–Yau cases, the Artin–Mazur height) through a single, explicit sequence of ideals with ht. They extend the criterion to graded complete intersections, enabling straightforward calculations in weighted/projective settings and deriving a Calabi–Yau height equality in this graded context. The paper then demonstrates significant applications, including the construction of Calabi–Yau varieties of arbitrarily high Artin–Mazur height over and explicit defining equations for quartic K3 surfaces over realizing a full range of heights, thereby highlighting both the theoretical depth and the algorithmic practicality of the approach. Overall, the results deepen the understanding of how Frobenius-related splitting properties govern formal-group heights and provide new tools to explore height phenomena in Calabi–Yau geometry in positive characteristic.

Abstract

Yobuko recently introduced the notion of quasi--splitting and quasi--split heights, which generalize and quantify the notion of Frobenius-splitting, and proved that quasi--split heights coincide with Artin-Mazur heights for Calabi-Yau varieties. In this paper, we prove Fedder type criteria for quasi--splittings of complete intersections, and in particular, obtain a simple formula to compute Artin-Mazur heights of Calabi-Yau hypersurfaces. As one of its applications, we prove that there exist Calabi-Yau varieties of arbitrarily high Artin-Mazur height over . We also give explicit defining equations of quartic K3 surfaces over realizing all the possible Artin-Mazur heights.
Paper Structure (26 sections, 39 theorems, 281 equations, 1 table)

This paper contains 26 sections, 39 theorems, 281 equations, 1 table.

Table of Contents

  1. Introduction
  2. Fedder type criteria for quasi-$F$-splitting
  3. Strategy of the proof of Theorem \ref{["Intro:thm:Fedder's criterion"]}
  4. Other applications of the main theorems
  5. Related work
  6. Notation and terminologies
  7. Preliminaries
  8. The ring of Witt vectors
  9. Quasi-$F$-splitting
  10. Regular sequence and colon ideals
  11. Structure of $\mathop{\mathrm{Hom}}\nolimits_{S}(Q_{S,n}, \mathcal{O}_S)$
  12. Construction of splitting maps
  13. Delta formula
  14. Fedder type criteria for quasi-$F$-splitting
  15. Proof.
  16. ...and 11 more sections

Key Result

Theorem A

Let $f\in S$ and $\theta$ an $S$-module homomorphism defined by We define an increasing sequence $\{I_n\}_n$ of ideals by $I_1:=(f^{p-1})$ and inductively. Then we have where $\inf \emptyset := \infty$. Furthermore, if $f$ is a homogeneous element and $N \geq 3$, then we have

Theorems & Definitions (112)

  • Theorem A: Fedder type criterion for quasi-$F$-splitting (see Theorem \ref{["thm:Fedder's criterion"]} and Corollary \ref{["cor:Fedder's criterion for quasi-F-splitting"]} for a more general statement)
  • Corollary B: cf. Corollary \ref{["cor:Fedder's criterion for quasi-F-splitting"]}
  • Theorem C: Calabi-Yau case of a Fedder type criterion for quasi-$F$-splitting (see Theorem \ref{["thm:Fedder's criterion for Calabi-Yau"]} for a more general statement)
  • Theorem D: Delta formula ( Remark \ref{['eg:explicit delta']} and Theorem \ref{['thm:delta formula']})
  • Lemma 2.1
  • proof
  • Definition 2.2
  • Remark 2.3
  • Lemma 2.4
  • proof
  • ...and 102 more