Algebraic geometry
On the homeomorphism type of some spaces of valuations
[Sur le type d'homéomorphisme des espaces de valuations]

Présenté par : Claire Voisin

de Felipe, Ana Belén1
1 Laboratoire de mathématiques UVSQ, bâtiment Fermat, 45, avenue des États-Unis, 78035 Versailles, France
Comptes Rendus. Mathématique, Tome 353 (2015) no. 6, pp. 541-544.

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.

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 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 non-Archimedean link of x in X.

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DOI : 10.1016/j.crma.2015.03.015
de Felipe, Ana Belén 1

1 Laboratoire de mathématiques UVSQ, bâtiment Fermat, 45, avenue des États-Unis, 78035 Versailles, France 
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de Felipe, Ana Belén. On the homeomorphism type of some spaces of valuations. Comptes Rendus. Mathématique, Tome 353 (2015) no. 6, pp. 541-544. doi : 10.1016/j.crma.2015.03.015. http://archive.numdam.org/articles/10.1016/j.crma.2015.03.015/

[1] Berkovich, V.G. Spectral Theory and Analytic Geometry over Non-Archimedean Fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990

[2] Cossart, V.; Piltant, O. Resolution of singularities of threefolds in positive characteristic. I.: Reduction to local uniformization on Artin–Schreier and purely inseparable coverings, J. Algebra, Volume 320 (2008) no. 3, pp. 1051-1082

[3] Fantini, L. Normalized Berkovich spaces and surface singularities, 2014 (arXiv preprint) | arXiv

[4] Favre, C. Holomorphic self-maps of singular rational surfaces, Publ. Mat., Volume 54 (2010) no. 2, pp. 389-432

[5] Favre, C.; Jonsson, M. The Valuative Tree, Lecture Notes in Mathematics, vol. 1853, Springer-Verlag, Berlin, 2004

[6] Herrera Govantes, F.J.; Olalla Acosta, M.A.; Spivakovsky, M.; Teissier, B. Extending a valuation centered in a local domain to the formal completion, Proc. Lond. Math. Soc., Volume 105 (2012) no. 3, pp. 571-621

[7] Hrushovski, E.; Loeser, F.; Poonen, B. Berkovich spaces embed in Euclidean spaces, Enseign. Math., Volume 60 (2014), pp. 273-293

[8] Mac Lane, S. Categories for the Working Mathematician, Graduate Texts in Mathematics, vol. 5, 1998

[9] Novacoski, J. Valuations centered at a two-dimensional regular local ring: infima and topologies, Segovia–El Escorial, 2011 (Congress Reports Series), European Math. Soc. Publishing House (Sept. 2014), pp. 389-403

[10] Pears, A.R. Dimension Theory of General Spaces, Cambridge University Press, Cambridge, UK, 1975

[11] Stallings, J.R. Topology of finite graphs, Invent. Math., Volume 71 (1983) no. 3, pp. 551-565

[12] Teissier, B. Overweight deformations of affine toric varieties and local uniformization, Segovia–El Escorial, Spain, 2011 (Congress Reports Series), European Mathematical Society Publishing House (Sept. 2014), pp. 474-565

[13] Thuillier, A. Géométrie toroidale et géométrie analytique non archimédienne. Application au type d'homotopie de certains schémas formels, Manuscr. Math., Volume 123 (2007) no. 4, pp. 381-451

[14] Zariski, O. The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc., Volume 50 (1944), pp. 683-691

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