Transcendental Hilbert-Kunz Multiplicities

Abstract

We prove that ordinary Hilbert-Kunz multiplicity can be transcendental. More precisely, over every uncountable algebraically closed field of characteristic $p>2$, there exists a normal standard graded domain $S$ and an $S_{+}$-primary homogeneous ideal $I\subseteq S$ such that $e_{\mathrm{HK}}(I{S}_{S_{+}})$ is transcendental.