Triangulations of non-archimedean curves, semi-stable reduction, and ramification

TL;DR

This paper investigates the relationship between minimal triangulations of non-archimedean curves and ramification extensions, providing new proofs and insights into semi-stable reduction and ramification behavior, especially in tame and wild cases.

Contribution

It establishes a divisibility relation between the minimal triangulation multiplicities and the degree of the Galois extension, offering a new proof of Saito's theorem and analyzing cases with marked points.

Findings

The least common multiple of triangulation multiplicities divides the extension degree.

If the multiplicity is prime to the residue characteristic, it equals the extension degree.

Examples show the limitations of extending Saito's theorem to wildly ramified cases.

Abstract

Let $K$ be a complete discretely valued field with algebraically closed residue field and let $\mathfrak C$ be a smooth projective and geometrically connected algebraic $K$-curve of genus $g$. Assume that $g\geq 2$, so that there exists a minimal finite Galois extension $L$ of $K$ such that $\mathfrak C_L$ admits a semi-stable model. In this paper, we study the extension $L|K$ in terms of the \emph{minimal triangulation} of $C$, a distinguished finite subset of the Berkovich analytification $C$ of $\mathfrak C$. We prove that the least common multiple $d$ of the multiplicities of the points of the minimal triangulation always divides the degree $[L:K]$. Moreover, if $d$ is prime to the residue characteristic of $K$, then we show that $d=[L:K]$, obtaining a new proof of a classical theorem of T. Saito. We then discuss curves with marked points, which allows us to prove analogous results in the case of elliptic curves, whose minimal triangulations we describe in full in the tame case. In the last section, we illustrate through several examples how our results explain the failure of the most natural extensions of Saito's theorem to the wildly ramified case.