Models of curves via Berkovich geometry

The theory of models of varieties is an important tool for topics such as deformation theory, moduli spaces, and degenerations. In the 1960s, Deligne and Mumford proved that any smooth projective curve C over a discretely valued field K has a semi-stable model after base-change to a finite Galois extension L|K. The question of determining such extension has been investigated ever since but has been settled only in the case where L|K is tamely ramified.
In this talk, I will present two results on the behaviour of models of curves under finite base-change. The first (joint with Lorenzo Fantini) exploits the geometry of the Berkovich analytification of C to describe the extension L|K in terms of regular models; the second (joint with Andrew Obus) investigates more in detail the case of potentially multiplicative reduction yielding new results in the case where L|K is wildly ramified.