Abstract
Let k be an algebraically closed field of characteristic p>0. We show that if X⊆Pkn is an equidimensional subscheme with Hilbert–Kunz multiplicity less than λ at all points x∈X, then for a general hyperplane H⊆Pkn, 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⊆Pkn 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.
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