Daniele Turchetti

Semi-stable models of curves and Berkovich geometry

The theory of models of curves sits at the intersection of number theory and geometry and it is rich of applications to Diophantine geometry, Galois representations, and cryptography among others.
In the 1960s, Deligne and Mumford proved that any smooth projective curve C over a discretely valued field K has a nice (i.e semi-stable) model after base-change to a finite Galois extension L|K.
The question of determining such extension has been extensively studied 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 base change. The first (joint with Lorenzo Fantini) exploits the geometry of the Berkovich analytification of C to relate certain regular models with the extension L|K; the second (joint with Andrew Obus) investigates more in detail the case of potentially multiplicative reduction, allowing us to find new results in the case where L|K is wildly ramified.