Article ID | Journal | Published Year | Pages | File Type |
---|---|---|---|---|
5778401 | Advances in Mathematics | 2017 | 35 Pages |
We show that the metric structure of morphisms f:YâX between quasi-smooth compact Berkovich curves over an algebraically closed field admits a finite combinatorial description. In particular, for a large enough skeleton Î=(ÎY,ÎX) of f, the sets Nf,â¥n of points of Y of multiplicity at least n in the fiber are radial around ÎY with the radius changing piecewise monomially along ÎY. In this case, for any interval l=[z,y]âY connecting a point z of type 1 to the skeleton, the restriction f|l gives rise to a profile piecewise monomial function Ïy:[0,1]â[0,1] that depends only on the type 2 point yâÎY. In particular, the metric structure of f is determined by Î and the family of the profile functions {Ïy} with yâÎY(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 Ïy coincides with the Herbrand function of H(y)/H(f(y)). This gives a curious geometric interpretation of the Herbrand function, which also applies to non-normal and even inseparable extensions.