Topology of spaces of valuations and geometry of singularities
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- by Ana Belén de Felipe PDF
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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
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Additional Information
- Ana Belén de Felipe
- Affiliation: Basque Center for Applied Mathematics (BCAM), Mazarredo 14, E48009 Bilbao, Basque Country, Spain
- MR Author ID: 1113309
- Email: adefelipe@bcamath.org
- Received by editor(s): July 5, 2016
- Received by editor(s) in revised form: July 7, 2017, and October 9, 2017
- Published electronically: September 18, 2018
- Additional Notes: This research was supported by ERCEA Consolidator Grant 615655 – NMST; the Basque Government through the BERC 2014–2017 program; the Spanish Ministry of Economy and Competitiveness MINECO; BCAM Severo Ochoa excellence accreditation SEV–2013–0323 and MTM2016-80659-P; and the ACIISI (with a cofinancing rate of 85% from ESF)
- © Copyright 2018 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 371 (2019), 3593-3626
- MSC (2010): Primary 14B05; Secondary 14E15
- DOI: https://doi.org/10.1090/tran/7441
- MathSciNet review: 3896123