A Riemann-Hurwitz Formula for Skeleta in Non-Archimedean Geometry

TL;DR

This paper develops a Riemann-Hurwitz type formula for skeleta in non-archimedean geometry, relating the genus of metric graph skeleta of curves under finite morphisms.

Contribution

It constructs compatible deformation retractions of analytified curves to skeleta and derives a formula relating their genera via the morphism.

Findings

Constructed compatible deformation retractions to skeleta.

Established a genus relation formula for skeleta under finite morphisms.

Provided a framework for calculating genus of skeleta in non-archimedean geometry.

Abstract

Let $\phi : C' \to C$ be a finite morphism between smooth, projective, irreducible curves defined over a non-archimedean valued, algebraically closed field $k$. This morphism induces a morphism between the analytifications of the curves. We will construct a compatible pair of deformation retractions of $C'^{an}$ and $C^{an}$ whose images $\Upsilon_{C'^{an}}$ and $\Upsilon_{C^{an}}$ are closed subspaces of $C'^{an}$ and $C^{an}$ which are homeomorphic to finite metric graphs. We refer to such closed subspaces as skeleta. In addition, the subspaces $C'^{an}$ and $C^{an}$ are such that their complements in the two analytifications decompose into the disjoint union of Berkovich open balls and annuli. To these skeleta we can associate a genus. The pair of compatible deformation retractions forces the morphism $\phi^{an}$ to restrict to a map $\Upsilon_{C'^{an}} \to \Upsilon_{C^{an}}$. We will study how the genus of $\Upsilon_{C'^{an}}$ can be calculated using the morphism $\phi^{an}: \Upsilon_{C'^{an}} \to \Upsilon_{C^{an}}$.