Publication date:
7 September 2017
Source:Advances in Mathematics, Volume 317
Author(s): Michael Temkin
We show that the metric structure of morphisms $f:Y\to X$ between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton $\mathrm{\Gamma }=({\mathrm{\Gamma }}_Y,{\mathrm{\Gamma }}_X)$ of f, the sets $N_{f,\ge n}$ of points of Y of multiplicity at least n in the fiber are radial around ${\mathrm{\Gamma }}_Y$ with the radius changing piecewise monomially along ${\mathrm{\Gamma }}_Y$. In this case, for any interval $l=[z,y]\subset Y$ connecting a point z of type 1 to the skeleton, the restriction $f{|}_l$ gives rise to a profile piecewise monomial function ${\phi }_y:[0,1]\to [0,1]$ that depends only on the type 2 point $y\in {\mathrm{\Gamma }}_Y$. In particular, the metric structure of f is determined by Γ and the family of the profile functions $\{{\phi }_y\}$ with $y\in {\mathrm{\Gamma }}_Y^{(2)}$. We prove that this family is piecewise monomial in y and naturally extends to the whole Y. In addition, we extend the classical theory of higher ramification groups to arbitrary real-valued fields and show that ${\phi }_y$ coincides with the Herbrand function of $\mathcal{H}(y)/\mathcal{H}(f(y))$. This gives a curious geometric interpretation of the Herbrand function, which also applies to non-normal and even inseparable extensions.