Bertini theorems for F-signature and Hilbert–Kunz multiplicity

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 \(x \in X \cap H\).