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

TL;DR

This paper develops criteria for quasi-$F$-splitting in algebraic geometry, enabling computation of Artin-Mazur heights for Calabi-Yau varieties and providing explicit examples over finite fields.

Contribution

It introduces Fedder type criteria for quasi-$F$-splitting of complete intersections and derives formulas for Artin-Mazur heights of Calabi-Yau hypersurfaces.

Findings

Existence of Calabi-Yau varieties with arbitrarily high Artin-Mazur height over $\\mathbb{F}_2$

Explicit equations for quartic K3 surfaces over $\\mathbb{F}_3$ with all Artin-Mazur heights

Simplified computation of Artin-Mazur heights using new criteria

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.