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

Let $X=P_{{\mathbb{F}}_p}^1-B$, 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: $$1\to {\pi }_1^t(X_{\overline{{\mathbb{F}}_q}})\to {\pi }_1^t(X_{{\mathbb{F}}_q})\to \mathrm{G}\mathrm{a}\mathrm{l}(\overline{{\mathbb{F}}_q}\mathrm{/}{\mathbb{F}}_q)\to 1.$$ This gives rise to an action of $\mathrm{G}\mathrm{a}\mathrm{l}(\overline{{\mathbb{F}}_q}\mathrm{/}{\mathbb{F}}_q)$ 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 $\mathrm{G}\mathrm{a}\mathrm{l}(\overline{{\mathbb{F}}_q}\mathrm{/}{\mathbb{F}}_q)$ 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 $X$.