1. Morphisms of Berkovich curves and the different function
- Author
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Cohen, Adina, Temkin, Michael, and Trushin, Dmitri
- Subjects
Mathematics - Algebraic Geometry - Abstract
Given a generically \'etale morphism $f\colon Y\to X$ of quasi-smooth Berkovich curves, we define a different function $\delta_f\colon Y\to[0,1]$ that measures the wildness of the topological ramification locus of $f$. This provides a new invariant for studying $f$, which cannot be obtained by the usual reduction techniques. We prove that $\delta_f$ is a piecewise monomial function satisfying a balancing condition at type 2 points analogous to the classical Riemann-Hurwitz formula, and show that $\delta_f$ can be used to explicitly construct the simultaneous skeletons of $X$ and $Y$. As an application, we use our results to completely describe the topological ramification locus of $f$ when its degree equals to the residue characteristic $p$., Comment: Final version, 49 pages, to appear in Adv.Math
- Published
- 2014