Comptes Rendus
Algebraic geometry
On the homeomorphism type of some spaces of valuations
Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 541-544.

Let X be an algebraic variety defined over an algebraically closed field. We study the fiber of the Riemann–Zariski space above a closed point xX. 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 non-Archimedean link of x in X.

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 xX. 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.

Published online:
DOI: 10.1016/j.crma.2015.03.015

Ana Belén de Felipe 1

1 Laboratoire de mathématiques UVSQ, bâtiment Fermat, 45, avenue des États-Unis, 78035 Versailles, France
     author = {Ana Bel\'en de Felipe},
     title = {On the homeomorphism type of some spaces of valuations},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {541--544},
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     year = {2015},
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     language = {en},
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Ana Belén de Felipe. On the homeomorphism type of some spaces of valuations. Comptes Rendus. Mathématique, Volume 353 (2015) no. 6, pp. 541-544. doi : 10.1016/j.crma.2015.03.015.

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