Hilbert-Kunz multiplicity and $F$-signature can disagree
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| Format: | Preprint |
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2025
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| _version_ | 1866911125752250368 |
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| author | Lee, Seungsu Pande, Suchitra Simpson, Austyn |
| author_facet | Lee, Seungsu Pande, Suchitra Simpson, Austyn |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2508_19985 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Hilbert-Kunz multiplicity and $F$-signature can disagree Lee, Seungsu Pande, Suchitra Simpson, Austyn Commutative Algebra Algebraic Geometry 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. |
| title | Hilbert-Kunz multiplicity and $F$-signature can disagree |
| topic | Commutative Algebra Algebraic Geometry |
| url | https://arxiv.org/abs/2508.19985 |