Hilbert-Kunz multiplicity of powers of ideals in dimension two
Alessandro De Stefani, Shreedevi K. Masuti, Maria Evelina Rossi, Jugal K. Verma
TL;DR
The paper investigates the asymptotic behavior of Hilbert-Kunz multiplicities for powers of ideals in dimension two, introducing a Ratliff–Rush closure-based invariant $ ilde{e}_{HK}$ to capture limiting behavior. It proves the existence of the secondary limits $L_2(I)$ and derives a polynomial-type expansion for $ ilde{e}_{HK}(I^n)$ in two dimensions, establishing equivalences and conditions under which Smirnov’s (Q1)-(C) hold, with explicit formulas in terms of reduction numbers and $ ilde{e}_{HK}$. The authors verify positive answers in several classes—numerically Roberts rings and parameter ideals in generalized Cohen–Macaulay rings—and relate (Q2) to tight closure via $ ilde{e}_{HK}$, including a transformation rule under finite maps. They also present potential counterexamples or evidence suggesting (Q2) may fail in general, supported by explicit computational examples in hypersurface rings with isolated singularities. The results deepen understanding of HK asymptotics, connect them to Ratliff–Rush closures and tight closure, and offer a framework for further exploring stability, reductions, and base changes in dimension two.
Abstract
We study the behavior of the Hilbert-Kunz multiplicity of powers of an ideal in a local ring. In dimension two, we provide answers to some problems raised by Smirnov, and give a criterion to answer one of his questions in terms of a "Ratliff-Rush version" of the Hilbert-Kunz multiplicity.