Stabilization indices of potentially Mumford curves

Abstract

Let $X$ be a smooth projective curve over a complete discretely valued field $K$. Let $L/K$ be the minimal extension such that $X \times_K L$ has a semi-stable model, and write $e(L/K)$ for the ramification index of $L/K$. Let $e(X)$ be the so-called ``stabilization index'' of $X$, defined by Halle and Nicaise as the lcm of the multiplicities of the ``principal'' irreducible components of a minimal regular snc-model of $X$. It is known that if $L/K$ is tame, then $e(X) = e(L/K)$. If one drops the tameness assumption, but instead assumes that $X$ has index one and potentially multiplicative reduction, Halle and Nicaise ask if the equality $e(X) = e(L/K)$ still holds. We prove that $e(X)$ divides $e(L/K)$ in this situation, but we give examples, in every residue characteristic, of $X$ with $K$-rational points and potentially multiplicative reduction such that $e(X) \neq e(L/K)$.