Metric uniformization of morphisms of Berkovich curves

TL;DR

This paper introduces a finite combinatorial framework to understand the metric structure of morphisms between Berkovich curves, revealing how ramification and fiber multiplicities are governed by piecewise monomial functions related to the Herbrand's function.

Contribution

It extends the metric uniformization theory of Berkovich curves, incorporating higher ramification groups and Herbrand's function into a geometric setting for arbitrary real-valued fields.

Findings

The metric structure is determined by a skeleton and profile functions.

Profile functions are piecewise monomial and extend to the entire curve.

Herbrand's function describes the ramification in this geometric context.

Abstract

We show that the metric structure of morphisms $f\colon Y\to X$ between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton $\Gamma=(\Gamma_Y,\Gamma_X)$ of $f$, the sets $N_{f,\ge n}$ of points of $Y$ of multiplicity at least $n$ in the fiber are radial around $\Gamma_Y$ with the radius changing piecewise monomially along $\Gamma_Y$. In this case, for any interval $l=[z,y]\subset Y$ connecting a rigid point $z$ to the skeleton, the restriction $f|_l$ gives rise to a $profile$ piecewise monomial function $\varphi_y\colon [0,1]\to[0,1]$ that depends only on the type 2 point $y\in\Gamma_Y$. In particular, the metric structure of $f$ is determined by $\Gamma$ and the family of the profile functions $\{\varphi_y\}$ with $y\in\Gamma_Y^{(2)}$. We prove that this family is piecewise monomial in $y$ and naturally extends to the whole $Y^{\mathrm{hyp}}$. In addition, we extend the theory of higher ramification groups to arbitrary real-valued fields and show that $\varphi_y$ coincides with the Herbrand's function of $\mathcal{H}(y)/\mathcal{H}(f(y))$. This gives a curious geometric interpretation of the Herbrand's function, which applies also to non-normal and even inseparable extensions.