Hilbert-Kunz multiplicity of two-dimensional local rings

Abstract

We study the behavior of Hilbert-Kunz multiplicity for powers of an ideal, especially the case of stable ideals and ideals in local rings of dimension $2$. We can characterize regular local rings by certain equality between Hilbert-Kunz multiplicity and usual multiplicity.

We show that rings with "minimal" Hilbert-Kunz multiplicity relative to usual multiplicity are "Veronese subrings" in dimension $2$.