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 $\phi^{ an}: \Upsilon_{ C'^{ an}} \to \Upsilon_{ C^{ an}}$.
Categories: math.AG