Topology of Spaces of Valuations and Geometry of Singularities

TL;DR

This paper investigates the topological structure of valuation spaces associated with algebraic varieties, revealing how their homeomorphism types depend on the nature of the point and the dimension, with connections to Berkovich geometry.

Contribution

It characterizes the homeomorphism types of valuation spaces for regular and singular points, linking them to resolution graphs and Berkovich geometry.

Findings

Homeomorphism type depends only on dimension for regular points.

For singular points on normal surfaces, it depends on the dual graph of a resolution.

Valuation space behavior matches that of the normalized non-Archimedean link.

Abstract

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