On the level of a Calabi-Yau hypersurface

TL;DR

This paper extends the concept of the level of hypersurfaces over finite fields to Calabi-Yau hypersurfaces, linking it to F-jumping exponents and Hartshorne-Speiser-Lyubeznik numbers, thus generalizing previous results from elliptic curves.

Contribution

It generalizes the notion of level from elliptic curves to Calabi-Yau hypersurfaces and relates it to F-jumping exponents and Hartshorne-Speiser-Lyubeznik numbers.

Findings

Level of Calabi-Yau hypersurfaces is characterized by F-jumping exponents.

Established a relation between level and Hartshorne-Speiser-Lyubeznik numbers.

Extended previous elliptic curve results to higher-dimensional Calabi-Yau cases.

Abstract

Boix-De Stefani-Vanzo defined the notion of level for a smooth projective hypersurface over a finite field in terms of the stabilisation of a chain of ideals previously considered by \`Alvarez-Montaner-Blickle-Lyubeznik, and showed that in the case of an elliptic curve the level is 1 if and only if it is ordinary and 2 otherwise. Here we extend their theorem to the case of Calabi-Yau hypersurfaces by relating their level to the $F$-jumping exponents of Blickle-Musta\c{t}\u{a}-Smith and the Hartshorne-Speiser-Lyubeznik numbers of Musta\c{t}\u{a}-Zhang.