Topology of spaces of valuations and geometry of singularities

de Felipe, Ana

doi:10.1090/tran/7441

Topology of spaces of valuations and geometry of singularities
HTML articles powered by AMS MathViewer

by Ana Belén de Felipe PDF

Trans. Amer. Math. Soc. 371 (2019), 3593-3626 Request permission

Abstract:

Given an algebraic variety $X$ defined over an algebraically closed field, we study the space $\mathrm {RZ}{(X,x)}$ 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 $\mathrm {RZ}{(X,x)}$ 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.

References

Vladimir G. Berkovich, Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33, American Mathematical Society, Providence, RI, 1990. MR 1070709, DOI 10.1090/surv/033
Włodzimierz J. Charatonik and Anne Dilks, On self-homeomorphic spaces, Topology Appl. 55 (1994), no. 3, 215–238. MR 1259506, DOI 10.1016/0166-8641(94)90038-8
Vincent Cossart and Olivier Piltant, Resolution of singularities of threefolds in positive characteristic. I. Reduction to local uniformization on Artin-Schreier and purely inseparable coverings, J. Algebra 320 (2008), no. 3, 1051–1082. MR 2427629, DOI 10.1016/j.jalgebra.2008.03.032
Steven Dale Cutkosky and Bernard Teissier, Semigroups of valuations on local rings, Michigan Math. J. 57 (2008), 173–193. Special volume in honor of Melvin Hochster. MR 2492447, DOI 10.1307/mmj/1220879403
Lorenzo Fantini, Normalized non-Archimedean links and surface singularities, C. R. Math. Acad. Sci. Paris 352 (2014), no. 9, 719–723 (English, with English and French summaries). MR 3258263, DOI 10.1016/j.crma.2014.06.010
L. Fantini, Normalized Berkovich spaces and surface singularities, arXiv:1412.4676 (2015).
Charles Favre, Holomorphic self-maps of singular rational surfaces, Publ. Mat. 54 (2010), no. 2, 389–432. MR 2675930, DOI 10.5565/PUBLMAT_{5}4210_{0}6
Charles Favre, Countability properties of some Berkovich spaces, Berkovich spaces and applications, Lecture Notes in Math., vol. 2119, Springer, Cham, 2015, pp. 119–132. MR 3330764, DOI 10.1007/978-3-319-11029-5_{3}
Charles Favre and Mattias Jonsson, The valuative tree, Lecture Notes in Mathematics, vol. 1853, Springer-Verlag, Berlin, 2004. MR 2097722, DOI 10.1007/b100262
Charles Favre and Mattias Jonsson, Valuative analysis of planar plurisubharmonic functions, Invent. Math. 162 (2005), no. 2, 271–311. MR 2199007, DOI 10.1007/s00222-005-0443-2
Charles Favre and Mattias Jonsson, Eigenvaluations, Ann. Sci. École Norm. Sup. (4) 40 (2007), no. 2, 309–349 (English, with English and French summaries). MR 2339287, DOI 10.1016/j.ansens.2007.01.002
A. Granja, The valuative tree of a two-dimensional regular local ring, Math. Res. Lett. 14 (2007), no. 1, 19–34. MR 2289617, DOI 10.4310/MRL.2007.v14.n1.a2
A. Grothendieck, Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Inst. Hautes Études Sci. Publ. Math. 24 (1965), 231 (French). MR 199181
F. J. Herrera Govantes, M. A. Olalla Acosta, M. Spivakovsky, and B. Teissier, Extending a valuation centred in a local domain to the formal completion, Proc. Lond. Math. Soc. (3) 105 (2012), no. 3, 571–621. MR 2974200, DOI 10.1112/plms/pds002
Ehud Hrushovski, François Loeser, and Bjorn Poonen, Berkovich spaces embed in Euclidean spaces, Enseign. Math. 60 (2014), no. 3-4, 273–292. MR 3342647, DOI 10.4171/LEM/60-3/4-4
Mattias Jonsson and Mircea Mustaţă, Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble) 62 (2012), no. 6, 2145–2209 (2013) (English, with English and French summaries). MR 3060755, DOI 10.5802/aif.2746
János Kollár, Simple normal crossing varieties with prescribed dual complex, Algebr. Geom. 1 (2014), no. 1, 57–68. MR 3234114, DOI 10.14231/AG-2014-004
Katarzyna Kuhlmann, The structure of spaces of $\Bbb {R}$-places of rational function fields over real closed fields, Rocky Mountain J. Math. 46 (2016), no. 2, 533–557. MR 3529082, DOI 10.1216/RMJ-2016-46-2-533
Josnei Novacoski, Valuations centered at a two-dimensional regular local domain: infima and topologies, Valuation theory in interaction, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2014, pp. 389–403. MR 3329042
A. R. Pears, Dimension theory of general spaces, Cambridge University Press, Cambridge, England-New York-Melbourne, 1975. MR 0394604
Jean-Pierre Serre, Trees, Springer-Verlag, Berlin-New York, 1980. Translated from the French by John Stillwell. MR 607504, DOI 10.1007/978-3-642-61856-7
John R. Stallings, Topology of finite graphs, Invent. Math. 71 (1983), no. 3, 551–565. MR 695906, DOI 10.1007/BF02095993
Bernard Teissier, Overweight deformations of affine toric varieties and local uniformization, Valuation theory in interaction, EMS Ser. Congr. Rep., Eur. Math. Soc., Zürich, 2014, pp. 474–565. MR 3329046
Amaury Thuillier, Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels, Manuscripta Math. 123 (2007), no. 4, 381–451 (French, with English summary). MR 2320738, DOI 10.1007/s00229-007-0094-2
Michel Vaquié, Valuations, Resolution of singularities (Obergurgl, 1997) Progr. Math., vol. 181, Birkhäuser, Basel, 2000, pp. 539–590 (French). MR 1748635
Oscar Zariski, The compactness of the Riemann manifold of an abstract field of algebraic functions, Bull. Amer. Math. Soc. 50 (1944), 683–691. MR 11573, DOI 10.1090/S0002-9904-1944-08206-2
Oscar Zariski and Pierre Samuel, Commutative algebra. Vol. II, Graduate Texts in Mathematics, Vol. 29, Springer-Verlag, New York-Heidelberg, 1975. Reprint of the 1960 edition. MR 0389876