Hilbert-Kunz multiplicity and $F$-signature can disagree

Abstract

We compute the $F$-signature function of the ample cone of any nontrivial ruled surface over ${\mathbb{P}}_k^1$ where $k$ is an algebraically closed field of prime characteristic. As an application, we construct a Noetherian $F$-finite strongly $F$-regular ring $R$ of prime characteristic admitting two maximal ideals ${\mathfrak{n}}_1,{\mathfrak{n}}_2\in \mathrm{Spec}R$ at which the Hilbert-Kunz multiplicity and $F$-signature measure different singularities; that is, $e_{\mathrm{HK}}(R_{{\mathfrak{n}}_1})<e_{\mathrm{HK}}(R_{{\mathfrak{n}}_2})$ and $s(R_{{\mathfrak{n}}_1})<s(R_{{\mathfrak{n}}_2})$. Our calculation of the $F$-signature for the Hirzebruch surfaces also corrects an inaccuracy in a preprint by different authors.

14 pages, 10 figures, comments welcome