Stabilization indices of potentially Mumford curves

TL;DR

This paper investigates the relationship between the stabilization index of a curve and the ramification index of its minimal extension for semi-stable models, proving divisibility but providing counterexamples to equality in certain cases.

Contribution

It proves that the stabilization index divides the ramification index for curves with index one and potentially multiplicative reduction, and presents counterexamples where equality fails.

Findings

e(X) divides e(L/K) in the specified setting

Counterexamples show e(X) ≠ e(L/K) despite having rational points

The equality holds under tame ramification but not in general

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 $e(X) \neq e(L/K)$.