Resolution of Nonsingularities for Mumford Curves

Abstract

Given a Mumford curve $X$ over ${\overline{\mathbf{Q}}}_p$, we show that for every semistable model $\mathcal{X}$ of $X$ and every closed point $x$ of this semistable model, there exists a finite étale cover $Y$ of $X$ such that every semistable model of $Y$ over $\mathcal{X}$ has a vertical component above $x$. We then give applications of this to the tempered fundamental group. In particular, we prove that two punctured Tate curves ${\overline{\mathbf{Q}}}_p$ with isomorphic tempered fundamental groups are isomorphic over ${\mathbf{Q}}_p$.