For equidimensional projective schemes over an algebraically closed field of characteristic p, having Hilbert-Kunz multiplicity below lambda everywhere is preserved by taking general hyperplane sections.
Bourbaki, Commutative algebra, Chapters 1-7 , Elements of Mathematics , Springer-Verlag, 1989
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Hilbert-Kunz Multiplicity of Fibers and Bertini Theorems
For equidimensional projective schemes over an algebraically closed field of characteristic p, having Hilbert-Kunz multiplicity below lambda everywhere is preserved by taking general hyperplane sections.