Metric uniformization of morphisms of Berkovich curves

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.