Topology of spaces of valuations and geometry of singularities

Abstract

Given an algebraic variety X X defined over an algebraically closed field, we study the space R Z ( X , x ) \mathrm {RZ}{(X,x)} consisting of all the valuations of the function field of X X which are centered in a closed point x x of X X . We concentrate on its homeomorphism type. We prove that, when x x is a regular point, this homeomorphism type only depends on the dimension of X X . If x x is a singular point of a normal surface, we show that it only depends on the dual graph of a good resolution of ( X , x ) (X,x) up to some precise equivalence. This is done by studying the relation between R Z ( X , x ) \mathrm {RZ}{(X,x)} and the normalized non-Archimedean link of x x in X X coming from the point of view of Berkovich geometry. We prove that their behavior is the same.