Bertini Theorems for $F$-signature and Hilbert-Kunz multiplicity

TL;DR

This paper establishes Bertini theorems for $F$-signature and Hilbert--Kunz multiplicity, showing that these invariants behave well under general hyperplane sections in normal quasi-projective varieties.

Contribution

It proves that $F$-signature and Hilbert--Kunz multiplicity satisfy Bertini-type theorems for general hyperplane sections in normal quasi-projective varieties.

Findings

Bertini theorems hold for $F$-signature and Hilbert--Kunz multiplicity.

General hyperplanes preserve bounds on $F$-signature and Hilbert--Kunz multiplicity.

Results apply to normal quasi-projective varieties with specified bounds.

Abstract

We show that Bertini theorems hold for $F$-signature and Hilbert--Kunz multiplicity. In particular, if $X \subseteq \mathbb{P}^n$ is normal and quasi-projective with $F$-signature greater than $\lambda$ (respectively the Hilbert--Kunz multiplicity is less than $\lambda$) at all points $x \in X$, then for a general hyperplane $H \subseteq \mathbb{P}^n$ the $F$-signature (respectively Hilbert--Kunz multiplicity) of $X \cap H$ is greater than $\lambda$ (respectively less than $\lambda$) at all points $x \in X \cap H$.