Normalized Berkovich spaces and surface singularities
Genormaliseerde Berkovich ruimten en singulariteiten van oppervlakken
Fantini, Lorenzo; R0252376
In this thesis we define normalized versions of Berkovich spaces over a trivially valued field k, obtained as quotients by the action of the group of positive real numbers which corresponds to rescaling semivaluations.The spaces that we obtain can be interpreted as non-archimedean models for the links of the singularities in varieties over k. We associate such a normalized space to any special formal k-scheme and study many of its properties.We prove a "normalized spaces version" of a classical theorem of Raynaud for non-archimedean analytic spaces, characterizing categorically the spaces that can be obtained in this way.This construction yields a locally ringed G-topological space, which we prove to be locally isomorphic to a non-archimedean analytic space over the field k((t)) with a t-adic valuation. Such a result allows us to approach the study of the birational geometry of k-varieties using techniques of non-archimedean geometry that are usually available only when working over a field with nontrivial valuation.In particular, we prove that the structure of the normalized non-archimedean links of surface singularities over an algebraically closed field k is analogous to the structure of non-archimedean analytic curves over k((t)).We deduce a characterization of the log essential valuations on a k-surface (i.e. those valuations whose center on every log resolution of the surface is a divisor) in terms of the local structure of the associated normalized space.In the last chapter of this thesis we reproduce a joint work with M. Cheung, J. Park and M. Ulirsch addressing a different problem: we study under which conditions a given tropical curve can be realized as the tropicalization of an algebraic curve whose skeleton is faithfully represented in the tropicalization.