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 $\mathbb{P}^1_k$ 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, $\operatorname{e}_{\operatorname{HK}}(R_{\mathfrak{n}_1})<\operatorname{e}_{\operatorname{HK}}(R_{\mathfrak{n}_2})$ 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.