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Normalized Berkovich spaces and surface singularities. (English) Zbl 1423.14171
Summary: We define normalized versions of Berkovich spaces over a trivially valued field \( k\), obtained as quotients by the action of \( \mathbb{R}_{>0}\) defined by rescaling semivaluations. We associate such a normalized space to any special formal \( k\)-scheme and prove an analogue of Raynaud’s theorem, characterizing categorically the spaces obtained in this way. This construction yields a locally ringed \( G\)-topological space, which we prove to be \( G\)-locally isomorphic to a Berkovich space over the field \( k((t))\) with a \( t\)-adic valuation. These spaces can be interpreted as non-archimedean models for the links of the singularities of \( k\)-varieties, and allow us to study the birational geometry of \( k\)-varieties using techniques of non-archimedean geometry available only when working over a field with non-trivial valuation. In particular, we prove that the structure of the normalized non-archimedean links of surface singularities over an algebraically closed field \( k\) is analogous to the structure of non-archimedean analytic curves over \( k((t))\) and deduce characterizations of the essential and of the log essential valuations, i.e., those valuations whose center on every resolution (respectively log resolution) of the given surface is a divisor.

MSC:
14G22 Rigid analytic geometry
14B05 Singularities in algebraic geometry
14E15 Global theory and resolution of singularities (algebro-geometric aspects)
14J17 Singularities of surfaces or higher-dimensional varieties
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[1] Abhyankar, Shreeram, Local uniformization on algebraic surfaces over ground fields of characteristic \(p\ne0\), Ann. of Math. (2), 63, 491-526 (1956) · Zbl 0108.16803
[2] Artin, Michael, Some numerical criteria for contractability of curves on algebraic surfaces, Amer. J. Math., 84, 485-496 (1962) · Zbl 0105.14404
[3] Artin, M., Algebraization of formal moduli. II. Existence of modifications, Ann. of Math. (2), 91, 88-135 (1970) · Zbl 0177.49003
[4] Artin, M., Coverings of the rational double points in characteristic \(p\). Complex analysis and algebraic geometry, 11-22 (1977), Iwanami Shoten, Tokyo
[5] Baker, Matthew; Payne, Sam; Rabinoff, Joseph, On the structure of non-Archimedean analytic curves. Tropical and non-Archimedean geometry, Contemp. Math. 605, 93-121 (2013), Amer. Math. Soc., Providence, RI · Zbl 1320.14040
[6] Ben-Bassat, Oren; Temkin, Michael, Berkovich spaces and tubular descent, Adv. Math., 234, 217-238 (2013) · Zbl 1288.14013
[7] Berkovich, Vladimir G., Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs 33, x+169 pp. (1990), American Mathematical Society, Providence, RI · Zbl 0715.14013
[8] Ber93 Vladimir G. Berkovich, \'Etale cohomology for non-Archimedean analytic spaces, Inst. Hautes \'Etudes Sci. Publ. Math. (1993), no. 78, 5-161. · Zbl 0804.32019
[9] Berkovich, Vladimir G., Vanishing cycles for formal schemes, Invent. Math., 115, 3, 539-571 (1994) · Zbl 0791.14008
[10] Berkovich, Vladimir G., Vanishing cycles for formal schemes. II, Invent. Math., 125, 2, 367-390 (1996) · Zbl 0852.14002
[11] Berthelot Pierre Berthelot, Cohomologie rigide et cohomologie rigide \`“a support propre, prepublication 96-03, Institut de Recherche Math\'”ematique Avanc\'ee, Rennes, 1996. · Zbl 0515.14015
[12] Bosch, Siegfried, Eine bemerkenswerte Eigenschaft der formellen Fasern affinoider R\"aume, Math. Ann., 229, 1, 25-45 (1977) · Zbl 0385.32008
[13] Bosch, Siegfried; L\"utkebohmert, Werner, Stable reduction and uniformization of abelian varieties. I, Math. Ann., 270, 3, 349-379 (1985) · Zbl 0554.14012
[14] Bosch, Siegfried; L\"utkebohmert, Werner, Formal and rigid geometry. I. Rigid spaces, Math. Ann., 295, 2, 291-317 (1993) · Zbl 0808.14017
[15] Bosch, Siegfried, Lectures on formal and rigid geometry, Lecture Notes in Mathematics 2105, viii+254 pp. (2014), Springer, Cham · Zbl 1314.14002
[16] Bosch, S.; G\"untzer, U.; Remmert, R., Non-Archimedean analysis, {\rm A systematic approach to rigid analytic geometry}, Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences] 261, xii+436 pp. (1984), Springer-Verlag, Berlin
[17] Boucksom, S.; de Fernex, T.; Favre, C.; Urbinati, S., Valuation spaces and multiplier ideals on singular varieties. Recent advances in algebraic geometry, London Math. Soc. Lecture Note Ser. 417, 29-51 (2015), Cambridge Univ. Press, Cambridge · Zbl 1330.14025
[18] Boucksom, S\'ebastien; Favre, Charles; Jonsson, Mattias, Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci., 44, 2, 449-494 (2008) · Zbl 1146.32017
[19] Bourbaki, Nicolas, Commutative algebra. Chapters 1-7, translated from the French, reprint of the 1989 English translation, Elements of Mathematics (Berlin), xxiv+625 pp. (1998), Springer-Verlag, Berlin · Zbl 0902.13001
[20] Cohen, I. S., On the structure and ideal theory of complete local rings, Trans. Amer. Math. Soc., 59, 54-106 (1946) · Zbl 0060.07001
[21] Fern\'andez de Bobadilla, Javier; Pereira, Mar\'\i a. Pe, The Nash problem for surfaces, Ann. of Math. (2), 176, 3, 2003-2029 (2012) · Zbl 1264.14049
[22] de Fernex, Tommaso, Three-dimensional counter-examples to the Nash problem, Compos. Math., 149, 9, 1519-1534 (2013) · Zbl 1285.14013
[23] de Fernex, Tommaso; Docampo, Roi, Terminal valuations and the Nash problem, Invent. Math., 203, 1, 303-331 (2016) · Zbl 1345.14020
[24] deFernexKollarXu12 Tommaso de Fernex, J\'anos Koll\'ar, and Chenyang Xu, The dual complex of singularities, preprint, arXiv:1212.1675 (2012). To appear in Proceedings of the conference in honor of Yujiro Kawamata’s 60th birthday, Advanced Studies in Pure Mathematics. · Zbl 1388.14107
[25] de Jong, A. J., Crystalline Dieudonn\'e module theory via formal and rigid geometry, Inst. Hautes \'Etudes Sci. Publ. Math., 82, 5-96 (1996) (1995) · Zbl 0864.14009
[26] Duc Antoine Ducros, La structure des courbes analytiques, book in preparation. The numbering in the text refers to the preliminary version of 12/02/2014, available at http://webusers.imj-prg.fr/\textasciitilde antoine.ducros/livre.html.
[27] Ducros, Antoine, Parties semi-alg\'ebriques d’une vari\'et\'e alg\'ebrique \(p\)-adique, Manuscripta Math., 111, 4, 513-528 (2003) · Zbl 1020.14017
[28] Ducros, Antoine, Espaces de Berkovich, polytopes, squelettes et th\'eorie des mod\`eles, Confluentes Math., 4, 4, 1250007, 57 pp. (2012) · Zbl 1263.14030
[29] Ducros, Antoine, Toute forme mod\'er\'ement ramifi\'ee d’un polydisque ouvert est triviale, Math. Z., 273, 1-2, 331-353 (2013) · Zbl 1264.14035
[30] Fantini, Lorenzo, Normalized non-Archimedean links and surface singularities, C. R. Math. Acad. Sci. Paris, 352, 9, 719-723 (2014) · Zbl 1319.14032
[31] Favre, Charles; Jonsson, Mattias, The valuative tree, Lecture Notes in Mathematics 1853, xiv+234 pp. (2004), Springer-Verlag, Berlin · Zbl 1064.14024
[32] Gabriel, P.; Zisman, M., Calculus of fractions and homotopy theory, Ergebnisse der Mathematik und ihrer Grenzgebiete, Band 35, x+168 pp. (1967), Springer-Verlag New York, Inc., New York · Zbl 0186.56802
[33] Grauert, Hans, \"Uber Modifikationen und exzeptionelle analytische Mengen, Math. Ann., 146, 331-368 (1962) · Zbl 0173.33004
[34] Grauert, Hans; Remmert, Reinhold, \"Uber die Methode der diskret bewerteten Ringe in der nicht-archimedischen Analysis, Invent. Math., 2, 87-133 (1966) · Zbl 0148.32401
[35] Grothendieck, A., \'El\'ements de g\'eom\'etrie alg\'ebrique. I. Le langage des sch\'emas, Inst. Hautes \'Etudes Sci. Publ. Math., 4, 228 pp. (1960) · Zbl 0118.36206
[36] Grothendieck, A., \'El\'ements de g\'eom\'etrie alg\'ebrique. III. \'Etude cohomologique des faisceaux coh\'erents. I, Inst. Hautes \'Etudes Sci. Publ. Math., 11, 167 pp. (1961)
[37] Grothendieck, A., \'El\'ements de g\'eom\'etrie alg\'ebrique. IV. \'Etude locale des sch\'emas et des morphismes de sch\'emas IV, Inst. Hautes \'Etudes Sci. Publ. Math., 32, 361 pp. (1967) · Zbl 0153.22301
[38] Hartshorne, Robin, Deformation theory, Graduate Texts in Mathematics 257, viii+234 pp. (2010), Springer, New York · Zbl 1186.14004
[39] Illusie, Luc, Grothendieck’s existence theorem in formal geometry. with a letter (in French) of Jean-Pierre Serre, Fundamental algebraic geometry, Math. Surveys Monogr. 123, 179-233 (2005), Amer. Math. Soc., Providence, RI
[40] Jonsson, Mattias; Musta\c t\u a, Mircea, Valuations and asymptotic invariants for sequences of ideals, Ann. Inst. Fourier (Grenoble), 62, 6, 2145-2209 (2013) (2012) · Zbl 1272.14016
[41] Kiehl, Reinhardt, Theorem A und Theorem B in der nichtarchimedischen Funktionentheorie, Invent. Math., 2, 256-273 (1967) · Zbl 0202.20201
[42] Kleiman, Steven L., Toward a numerical theory of ampleness, Ann. of Math. (2), 84, 293-344 (1966) · Zbl 0146.17001
[43] Knutson, Donald, Algebraic spaces, Lecture Notes in Mathematics, Vol. 203, vi+261 pp. (1971), Springer-Verlag, Berlin-New York · Zbl 0221.14001
[44] Koll\'ar, J\'anos, Links of complex analytic singularities. Surveys in differential geometry. Geometry and topology, Surv. Differ. Geom. 18, 157-193 (2013), Int. Press, Somerville, MA · Zbl 1318.32033
[45] Lipman, Joseph, Rational singularities, with applications to algebraic surfaces and unique factorization, Inst. Hautes \'Etudes Sci. Publ. Math., 36, 195-279 (1969) · Zbl 0181.48903
[46] Liu, Qing, Sur les espaces de Stein quasi-compacts en g\'eom\'etrie rigide, Tohoku Math. J. (2), 42, 3, 289-306 (1990) · Zbl 0716.32023
[47] Martin, Florent, Analytic functions on tubes of nonarchimedean analytic spaces, {\rm with an appendix by Christian Kappen and Martin}, Algebra Number Theory, 11, 3, 657-683 (2017) · Zbl 1401.14129
[48] Nash, John F., Jr., Arc structure of singularities, {\rm A celebration of John F. Nash, Jr.}, Duke Math. J., 81, 1, 31-38 (1996) (1995) · Zbl 0880.14010
[49] Nicaise, Johannes, Formal and rigid geometry: an intuitive introduction and some applications, Enseign. Math. (2), 54, 3-4, 213-249 (2008) · Zbl 1172.14001
[50] Nicaise, Johannes, A trace formula for rigid varieties, and motivic Weil generating series for formal schemes, Math. Ann., 343, 2, 285-349 (2009) · Zbl 1177.14050
[51] Nicaise, Johannes, Singular cohomology of the analytic Milnor fiber, and mixed Hodge structure on the nearby cohomology, J. Algebraic Geom., 20, 2, 199-237 (2011) · Zbl 1226.14008
[52] Payne, Sam, Topology of nonarchimedean analytic spaces and relations to complex algebraic geometry, Bull. Amer. Math. Soc. (N.S.), 52, 2, 223-247 (2015) · Zbl 1317.32046
[53] P\'erez Rodr\'\i guez, Marta, Basic deformation theory of smooth formal schemes, J. Pure Appl. Algebra, 212, 11, 2381-2388 (2008) · Zbl 1151.14007
[54] Raynaud, Michel, G\'eom\'etrie analytique rigide d’apr\`“es Tate, Kiehl,\( \cdots \). Table Ronde d”Analyse non archim\'edienne, Paris, 1972, 319-327 (1974), Bull. Soc. Math. France, M\'em. No. 39-40, Soc. Math. France, Paris · Zbl 0299.14003
[55] Temkin, M., A new proof of the Gerritzen-Grauert theorem, Math. Ann., 333, 2, 261-269 (2005) · Zbl 1080.32021
[56] Temkin, Michael, Introduction to Berkovich analytic spaces. Berkovich spaces and applications, Lecture Notes in Math. 2119, 3-66 (2015), Springer, Cham · Zbl 1317.14054
[57] Thuillier, Amaury, G\'eom\'etrie toro\`“\i dale et g\'”eom\'etrie analytique non archim\'edienne. Application au type d’homotopie de certains sch\'emas formels, Manuscripta Math., 123, 4, 381-451 (2007) · Zbl 1134.14018
[58] Valabrega, Paolo, On the excellent property for power series rings over polynomial rings, J. Math. Kyoto Univ., 15, 2, 387-395 (1975) · Zbl 0306.13011
[59] Valabrega, Paolo, A few theorems on completion of excellent rings, Nagoya Math. J., 61, 127-133 (1976) · Zbl 0319.13008
[60] Vaqui\'e, Michel, Valuations. Resolution of singularities, Obergurgl, 1997, Progr. Math. 181, 539-590 (2000), Birkh\"auser, Basel · Zbl 1003.13001
[61] Zariski, Oscar, The reduction of the singularities of an algebraic surface, Ann. of Math. (2), 40, 639-689 (1939) · JFM 65.1399.03
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