Bertini theorems for F-signature and Hilbert–Kunz multiplicity

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