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.

Let I be an ideal of the ring of formal power series K[ [x, y]] with coefficients in an algebraically closed field K of arbitrary characteristic. Let Φ denote the set of all parametrizations ϕ = (where ϕ = (0, 0) and ϕ(0, 0) = (0, 0). The purpose of this paper is to investigate the invariantcalled the Lojasiewicz exponent of I. Our main result states that for the ideals I of finite codimension the Lojasiewicz exponent L 0 (I) is a Farey number i.e. an integer or a rational number of the form N + b a , where a, b, N are integers such that 0 < b < a < N .

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Presented by Claire VoisinLet X be an algebraic variety defined over an algebraically closed field. We study the fiber of the Riemann-Zariski space above a closed point x ∈ X. If x is regular, we prove that its 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. Both results also hold for the normalized nonArchimedean link of x in X.© 2015 Académie des sciences. Published by Elsevier Masson SAS. All rights reserved. r é s u m é Soit X une variété algébrique définie sur un corps algébriquement clos. On étudie la fibre de l'espace de Riemann-Zariski au-dessus d'un point fermé x ∈ X. Si x est régulier, on démontre que son type d'homéomorphisme ne dépend que de la dimension de X. Si x est un point singulier d'une surface normale, on démontre qu'il ne dépend que de la classe du graphe d'une bonne résolution de (X, x) modulo une relation d'équivalence précise. Ces deux résultats sont aussi vrais pour l'entrelac non archimédien normalisé de x dans X.