# Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems

## TL;DR

This paper proves a Bertini-type theorem for Hilbert--Kunz multiplicity, showing that general hyperplane sections preserve bounds on multiplicity for equidimensional schemes in projective space over a field of positive characteristic.

## Contribution

It establishes a new Bertini theorem for Hilbert--Kunz multiplicity, extending previous results to non-normal schemes and providing generalized uniform estimates for fibers.

## Key findings

- Hilbert--Kunz multiplicity remains bounded under general hyperplane sections.
- The results generalize prior theorems to broader classes of schemes.
- Provides new uniform estimates for Hilbert--Kunz multiplicities of fibers.

## Abstract

Let $k$ be an algebraically closed field of characteristic $p > 0$. We show that if $X\subseteq\mathbb{P}^n_k$ is an equidimensional subscheme with Hilbert--Kunz multiplicity less than $\lambda$ at all points $x\in X$, then for a general hyperplane $H\subseteq\mathbb{P}^n_k$, the Hilbert--Kunz multiplicity of $X\cap H$ is less than $\lambda$ at all points $x\in X\cap H$. This answers a conjecture and generalizes a result of Carvajal-Rojas, Schwede and Tucker, whose conclusion is the same as ours when $X\subseteq\mathbb{P}^n_k$ is normal. In the process, we substantially generalize certain uniform estimates on Hilbert--Kunz multiplicities of fibers of maps obtained by the aforementioned authors that should be of independent interest.