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arXiv · 1711.03909

Topology of Spaces of Valuations and Geometry of Singularities

Ana B. de Felipe
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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.

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Ana B. de Felipe. 2017-11-10. Topology of Spaces of Valuations and Geometry of Singularities. https://doi.org/10.1090/tran%2F7441

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