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Transactions of the American Mathematical Society

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



Normalized Berkovich spaces and surface singularities

Author: Lorenzo Fantini
Journal: Trans. Amer. Math. Soc.
MSC (2010): Primary 14E15, 14G22; Secondary 14J17
Published electronically: May 9, 2018
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Abstract: 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).
Article copyright: © Copyright 2018 by the author

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