On the number of connected components of the ramification locus of a morphism of Berkovich curves

TL;DR

This paper establishes upper bounds on the number of connected components of the ramification locus for finite morphisms of quasi-smooth non-archimedean analytic curves, linking it to topological invariants of the source curve.

Contribution

It provides new bounds for the ramification locus components based on topological invariants, advancing understanding of non-archimedean analytic curve morphisms.

Findings

Upper bounds depend on topological genus, boundary points, and open ends.

Results apply to curves admitting finite triangulations.

Enhances understanding of ramification in non-archimedean geometry.

Abstract

Let $k$ be a complete, nontrivially valued non-archimedean field. Given a finite morphism of quasi-smooth $k$-analytic curves that admit finite triangulations, we provide upper bounds for the number of connected components of the ramification locus in terms of topological invariants of the source curve such as its topological genus, the number of points in the boundary and the number of open ends.