标题： Hilbert-Kunz 乘数和$F$-签名可能会不一致 显示英文标题

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

摘要： 我们计算任何在$\mathbb{P}^1_k$上的非平凡可规曲面的$F$-签名函数，其中$k$是一个素数特征的代数闭域。 作为应用，我们构造了一个诺特的$F$-有限的强$F$-正则环$R$，其素特征承认两个极大理想$\mathfrak{n}_1,\mathfrak{n}_2\in \mathrm{Spec} R$，在这些极大理想处，Hilbert-Kunz 乘数和$F$-签名度量不同的奇点；即，$\operatorname{e}_{\operatorname{HK}}(R_{\mathfrak{n}_1})<\operatorname{e}_{\operatorname{HK}}(R_{\mathfrak{n}_2})$和$s(R_{\mathfrak{n}_1})<s(R_{\mathfrak{n}_2})$。 我们对Hirzebruch曲面的$F$-符号的计算也修正了其他作者一篇预印本中的一个不准确之处。 显示英文摘要

摘要： 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.