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 $X$.
Categories: math.AG math.AT