Bertini theorems for F-singularities

TL;DR

This paper establishes Bertini-type theorems for strongly F-regular and F-pure singularities, but shows F-injective singularities do not satisfy such properties, advancing understanding of singularity behavior under hyperplane sections.

Contribution

It proves Bertini theorems for certain F-singularities and demonstrates limitations for F-injective singularities, extending the theoretical framework of singularity behavior.

Findings

Strongly F-regular and F-pure singularities satisfy Bertini theorems.

F-injective singularities do not satisfy basic Bertini properties.

The work builds on and extends existing frameworks for singularity analysis.

Abstract

We prove that strongly F-regular and F-pure singularities satisfy Bertini-type theorems (including in the context of pairs) by building upon a framework of Cumino, Greco and Manaresi (compare with the work of Jouanolou and Spreafico). We also prove that F-injective singularities fail to satisfy even the most basic Bertini-type results.