Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems

Abstract

Let $k$ be an algebraically closed field of characteristic $p>0$. We show that if $X\subseteq {\mathbb{P}}_k^n$ is an equidimensional subscheme with Hilbert--Kunz multiplicity less than $\lambda$ at all points $x\in X$, then for a general hyperplane $H\subseteq {\mathbb{P}}_k^n$, 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 $X\subseteq {\mathbb{P}}_k^n$ 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.

Comments welcome; v2 has minor edits and corrections