A Seifert-van Kampen Theorem and the Frobenius Action on Tame Fundamental Groups

Abstract

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: \[1\to \pi_1^t(X_{\overline{\mathbb{F}_q}})\to \pi_1^t(X_{\mathbb{F}_q})\to \mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)\to 1.\] This gives rise to an action of $\mathrm{Gal}(\overline{\mathbb{F}_q}/\mathbb{F}_q)$ on $\pi_1^t(X_{\overline{\mathbb{F}_q}})$ 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 $\pi_1^t(X_{\overline{\mathbb{F}_q}})$ assigned to each geometric point of $B_{\overline{\mathbb{F}_q}}$. Based on this, we give a purely algebraic computation of $\pi_1^t(X_{\overline{\mathbb{F}_q}})$, and thereby obtain an explicit description of the tame fundamental group of $X$.