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February 2022 Tropical geometry over the tropical hyperfield
Oliver Lorscheid
Rocky Mountain J. Math. 52(1): 189-222 (February 2022). DOI: 10.1216/rmj.2022.52.189

Abstract

We merge ideas around the tropical hyperfield with the theory of ordered blueprints to give a new formulation of tropical scheme theory. The key insight is that a nonarchimedean absolute value can be considered as a morphism into the tropical hyperfield. In turn, ordered blueprints make it possible to consider the base change of a classical variety to the tropical hyperfield. We call this base change the scheme theoretic tropicalization of the classical variety.

Our first main result describes the Berkovich analytification and the tropicalization of a classical variety as sets of rational points of scheme theoretic tropicalizations, including a characterization of the respective topologies. Our second main result shows that the Giansiracusa bend relations can be derived by a natural construction from the scheme theoretic tropicalization.

Citation

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Oliver Lorscheid. "Tropical geometry over the tropical hyperfield." Rocky Mountain J. Math. 52 (1) 189 - 222, February 2022. https://doi.org/10.1216/rmj.2022.52.189

Information

Received: 22 September 2020; Revised: 26 May 2021; Accepted: 7 June 2021; Published: February 2022
First available in Project Euclid: 19 April 2022

Digital Object Identifier: 10.1216/rmj.2022.52.189

Subjects:
Primary: 06F05 , 14T10 , 16Y60
Secondary: 08A30 , 12J20 , 14G22

Keywords: Berkovich spaces , Giansiracusa tropicalization , ordered blueprint , tropical hyperfield , tropical scheme theory

Rights: Copyright © 2022 Rocky Mountain Mathematics Consortium

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Vol.52 • No. 1 • February 2022
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