A unifying approach to tropicalization
Oliver Lorscheid

TL;DR
This paper introduces ordered blueprints and schemes to unify and enhance tropicalization methods, providing a schematic framework that generalizes existing approaches and connects various tropical and analytic geometries.
Contribution
It develops a unifying schematic approach to tropicalization using ordered blueprints, improving existing methods and incorporating various tropical and analytic geometries.
Findings
Recover and improve existing tropicalization approaches
Represent Berkovich and Kajiwara-Payne tropicalizations as rational points
Generalize tropicalizations to nontrivial valuations with schematic structure
Abstract
In this paper, we introduce ordered blueprints and ordered blue schemes, which serve as a common language for the different approaches to tropicalizations and which enhances tropical varieties with a schematic structure. As an abstract concept, we consider a tropicalization as a moduli problem about extensions of a given valuation between ordered blueprints and . If is idempotent, then we show that a generalization of the Giansiracusa bend relation leads to a representing object for the tropicalization, and that it has yet another interpretation in terms of a base change along . We call such a representing object a scheme theoretic tropicalization. This theory recovers and improves other approaches to tropicalizations as we explain with care in the second part of this text. The Berkovich analytification and the Kajiwara-Payne tropicalization appear as…
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Taxonomy
TopicsPolynomial and algebraic computation
