Prime-to-$p$ étale fundamental groups of punctured projective lines over strictly Henselian fields

Abstract: Let $K$ be the fraction field of a strictly Henselian DVR of characteristic $p \geq 0$ with algebraic closure $\bar{K}$, and let $\alpha_{1}, ..., \alpha_{d} \in \mathbb{P}{K}^{1}(K)$. In this paper, we give explicit generators and relations for the prime-to-$p$ \'etale fundamental group of $\mathbb{P}_K^1\smallsetminus{\alpha_1,...,\alpha_d}$ that depend (solely) on their intersection behavior. This is done by a comparison theorem that relates this situation to a topological one. Namely, let $a{1}, ..., a_{d}$ be distinct power series in $\mathbb{C}[[x]]$ with the same intersection behavior as the $\alpha_i$'s, converging on an open disk centered at $0$, and choose a point $z_{0} \neq 0$ lying in this open disk. We compare the natural action of $\mathrm{Gal}(K)$ on the prime-to-$p$ \'etale fundamental group of $\mathbb{P}{\bar{K}} \smallsetminus {\alpha{1}, ..., \alpha_{d}}$ to the topological action of looping $z_0$ around the origin on the fundamental group of $\mathbb{P}{\mathbb{C}}^1 \smallsetminus {a_1(z_0),...,a_d(z_0)}$. This latter action is, in turn, interpreted in terms of Dehn twists. A corollary of this result is that every prime-to-$p$ $G$-Galois cover of $\mathbb{P}{\bar K}^1 \smallsetminus {\alpha_1,...,\alpha_d}$ satisfies that its field of moduli (as a $G$-Galois cover) has degree over $K$ dividing the exponent of $G / Z(G)$.