A Seifert-van Kampen Theorem and the Frobenius Action on Tame Fundamental Groups
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866918156967084032 |
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| author | Yao, Yuxiang |
| author_facet | Yao, Yuxiang |
| contents | Let $X = P^1_{\mathbb{F}_p}-B$, where $B$ is a divisor with $n$ distinct geometric points, and view $X$ as a $\mathbb{F}_q$-variety with $q = p^r$ for some $r$, we then obtain a short exact sequence of tame fundamental groups:
\[1\to π_1^t(X_{\overline{\mathbb{F}_q}})\to π_1^t(X_{\mathbb{F}_q})\to \mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)\to 1.\]
This gives rise to an action of $\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ on $π_1^t(X_{\overline{\mathbb{F}_q}})$ once an $\mathbb{F}_q$-point in $X_{\mathbb{F}_q}$ is fixed. Using Harbater's formal patching, we prove a version of the Seifert-van Kampen theorem, which further yields a purely algebraic description of the action of $\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ on the $n$ generators of $π_1^t(X_{\overline{\mathbb{F}_q}})$ assigned to each geometric point of $B_{\overline{\mathbb{F}_q}}$. Based on this, we give a purely algebraic computation of $π_1^t(X_{\overline{\mathbb{F}_q}})$, and thereby obtain an explicit description of the tame fundamental group of $X$. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2509_17551 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | A Seifert-van Kampen Theorem and the Frobenius Action on Tame Fundamental Groups Yao, Yuxiang Algebraic Geometry Algebraic Topology Let $X = P^1_{\mathbb{F}_p}-B$, where $B$ is a divisor with $n$ distinct geometric points, and view $X$ as a $\mathbb{F}_q$-variety with $q = p^r$ for some $r$, we then obtain a short exact sequence of tame fundamental groups: \[1\to π_1^t(X_{\overline{\mathbb{F}_q}})\to π_1^t(X_{\mathbb{F}_q})\to \mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)\to 1.\] This gives rise to an action of $\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ on $π_1^t(X_{\overline{\mathbb{F}_q}})$ once an $\mathbb{F}_q$-point in $X_{\mathbb{F}_q}$ is fixed. Using Harbater's formal patching, we prove a version of the Seifert-van Kampen theorem, which further yields a purely algebraic description of the action of $\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ on the $n$ generators of $π_1^t(X_{\overline{\mathbb{F}_q}})$ assigned to each geometric point of $B_{\overline{\mathbb{F}_q}}$. Based on this, we give a purely algebraic computation of $π_1^t(X_{\overline{\mathbb{F}_q}})$, and thereby obtain an explicit description of the tame fundamental group of $X$. |
| title | A Seifert-van Kampen Theorem and the Frobenius Action on Tame Fundamental Groups |
| topic | Algebraic Geometry Algebraic Topology |
| url | https://arxiv.org/abs/2509.17551 |