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Transactions of the American Mathematical Society

Published by the American Mathematical Society since 1900, Transactions of the American Mathematical Society is devoted to longer research articles in all areas of pure and applied mathematics.

ISSN 1088-6850 (online) ISSN 0002-9947 (print)

The 2020 MCQ for Transactions of the American Mathematical Society is 1.48.

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Normalized Berkovich spaces and surface singularities
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by Lorenzo Fantini PDF
Trans. Amer. Math. Soc. 370 (2018), 7815-7859


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.
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Additional Information
  • Lorenzo Fantini
  • Affiliation: Institut Mathématique de Jussieu, Université Pierre et Marie Curie, 75252 Paris, France
  • Address at time of publication: CNRS, Sorbonne Université, Université Paris Diderot, Institut de Mathématiques de Jussieu-Paris Rive Gauche, IMJ-PRG, F-75005 Paris, France
  • Received by editor(s): April 17, 2015
  • Received by editor(s) in revised form: October 2, 2016, February 5, 2017, and February 9, 2017
  • Published electronically: May 9, 2018
  • Additional Notes: During the preparation of this work, the author’s research was supported by the Fund for Scientific Research - Flanders (grant G.0415.10) and the European Research Council (Starting Grant project “Nonarcomp” no.307856).
  • © Copyright 2018 by the author
  • Journal: Trans. Amer. Math. Soc. 370 (2018), 7815-7859
  • MSC (2010): Primary 14E15, 14G22; Secondary 14J17
  • DOI:
  • MathSciNet review: 3852450