Hilbert-Kunz multiplicity of three-dimensional local rings

Abstract

In this paper, we investigate the lower bound $s_{HK}(p, d)$ of Hilbert-Kunz multiplicities for non-regular unmixed local rings of Krull dimension $d$ containing a field of characteristic $p>0$. Especially, we focus on three-dimensional local rings. In fact, as a main result, we will prove that $s_{HK}(p, 3) = 4/3$ and that a three-dimensional complete local ring of Hilbert-Kunz multiplicity $4/3$ is isomorphic to the non-degenerate quadric hypersurface $k[[X, Y, Z, W]]/(X^{2}+Y^{2}+Z^{2}+W^{2})$ under mild conditions.

Furthermore, we pose a generalization of the main theorem to the case of $\dim A \ge 4$ as a conjecture, and show that it is also true in case $\dim A = 4$ using the similar method as in the proof of the main theorem.