Hilbert–Kunz multiplicity of fibers and Bertini theorems

Let k be an algebraically closed field of characteristic p > 0 . We show that if X ⊆ P k n is an equidimensional subscheme with Hilbert–Kunz multiplicity less than λ at all points x ∈ X , then for a general hyperplane H ⊆ P k n , the Hilbert–Kunz multiplicity of X ∩ H is less than λ at all points x ∈ X ∩ H . This answers a conjecture and generalizes a result of Carvajal-Rojas, Schwede and Tucker, whose conclusion is the same as ours when X ⊆ 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.