On the level of a Calabi-Yau hypersurface

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 Àlvarez-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-Mustaţă-Smith and the Hartshorne-Speiser-Lyubeznik numbers of Mustaţă-Zhang.

7 pages. This version: (old) proposition 7 missing an hypothesis and proof was incorrect, but it actually follows for CYs from the proof of (old) proposition 17 so an explicit F-splitting is unnecessary; the paper is consequently reorganised. Some other relatively minor errata/notation corrected. Main result is unaffected. Thanks to Adrian Langer for a number of useful comments