Algebraic and Combinatorial Aspects of Tropical Geometry

Popescu-Pampu, Patrick; Stepanov, D. A.; Dickenstein, Alicia; Feichtner, Eva-Maria; Itenberg, Ilia; Brugallé, Erwan; Cueto, María Angélica

doi:10.1090/conm/589

Cited by 12 publications

(27 citation statements)

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“…Remark 3.1.1. The idea of generalizing the ambient spaces for tropicalization from toric varieties to F 1 -schemes first appeared in [PPS13], although they worked only with set-theoretic tropicalization.…”

Section: 1mentioning

confidence: 99%

The universal tropicalization and the Berkovich analytification

Giansiracusa¹,

Giansiracusa²

2014

Preprint

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Given an integral scheme X over a non-archimedean valued field k, we construct a universal closed embedding of X into a k-scheme equipped with a model over the field with one element F 1 (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of X by [GG13], and we show that the set-theoretic tropicalization of X with respect to this universal embedding is the Berkovich analytification X an . Moreover, using the scheme-theoretic tropicalization of [GG13], we obtain a tropical scheme Trop univ (X) whose T-points give the analytification and that canonically maps to all other scheme-theoretic tropicalizations of X. This makes precise the idea that the Berkovich analytification is the universal tropicalization. When X = Spec A is affine, we show that Trop univ (X) is the limit of the tropicalizations of X with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that Trop univ (X) represents the moduli functor of valuations on X, and when X = Spec A is affine there is a universal valuation on A taking values in the semiring of regular functions on the universal tropicalization.

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Section: 1mentioning

confidence: 99%

The universal tropicalization and the Berkovich analytification

Giansiracusa¹,

Giansiracusa²

2014

Preprint

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“…Our approach to the construction of trop X is a global version of the local tropicalization map defined by Popescu-Pampu and Stepanov in [43,Section 6]. Fix a morphism α : P → A from a monoid P into the multiplicative monoid of an algebra A of finite type over k (or more generally into the quotient A/A * ) and set X = Spec A.…”

Section: The Main Idea Of Our Constructionmentioning

confidence: 99%

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

Ulirsch

2017

Proc. London Math. Soc.

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Abstract. The purpose of this article is to develop foundational techniques from logarithmic geometry in order to define a functorial tropicalization map for fine and saturated logarithmic schemes in the case of constant coefficients. Our approach crucially uses the theory of fans in the sense of K. Kato and generalizes Thuillier's retraction map onto the non-Archimedean skeleton in the toroidal case. For the convenience of the reader many examples as well as an introductory treatment of the theory of Kato fans are included.

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“…Thus, on the relative interior of each cone, the image of Z identified with the tropicalization of a subvariety of a toric variety. The fundamental theorem of tropical geometry implies that the image of Z in the cone above is polyhedral [36, Theorem 1.1], see also [8,21,27].…”

Section: Polyhedralitymentioning

confidence: 99%

Skeletons of stable maps II: superabundant geometries

Ranganathan

2017

Res Math Sci

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We implement new techniques involving Artin fans to study the realizability of tropical stable maps in superabundant combinatorial types. Our approach is to understand the skeleton of a fundamental object in logarithmic Gromov-Witten theory-the stack of prestable maps to the Artin fan. This is used to examine the structure of the locus of realizable tropical curves and derive three principal consequences. First, we prove a realizability theorem for limits of families of tropical stable maps. Second, we extend the sufficiency of Speyer's well-spacedness condition to the case of curves with good reduction. Finally, we demonstrate the existence of liftable genus 1 superabundant tropical curves that violate the well-spacedness condition. BackgroundCentral to the application of tropical techniques to questions in algebraic geometry are so-called lifting theorems. Given a "synthetic" tropical object, such as a weighted balanced polyhedral complex, one must understand whether this object is the tropicalization of an algebraic variety. We deal in this paper with the case of curves. The tropical lifting question in this setting asks, when does an embedded tropical curve in R n arise as the tropicalization of an algebraic curve in a torus over a nonarchimedean field? This question becomes highly nontrivial in the so-called superabundant case and has been the primary obstacle to the application of tropical curve counting techniques in high genus settings. A tropical curve in R n encodes the combinatorial data in a degenerate logarithmic stable map to a toric variety. If the tropical curve is superabundant, i.e., if the tropical deformation space is larger than expected, the obstruction group of this degenerate logarithmic map is nonzero. As a result, such a map may not deform and the tropical curve may fail to be realizable. See Sect. 2.2 for a precise definition of superabundance. The earliest realization theorems for superabundant curves are due to Speyer, who observed a subtle combinatorial condition guaranteeing the realizability of superabundant genus 1 tropical curves [31]. While there has been substantial additional work in the intervening years, the general question remains mysterious [7,15,19,[23][24][25]29,34].In this note, we use recent technical breakthroughs in nonarchimedean geometry and the theory of logarithmic maps to provide a conceptual framework in which the realizability question may be approached. To demonstrate the efficacy of this framework, we use it to give simple proofs of three new results for lifting tropical curves. The same frame- 0123456789().,-: vol

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Algebraic and Combinatorial Aspects of Tropical Geometry

Cited by 12 publications

References 0 publications

The universal tropicalization and the Berkovich analytification

The universal tropicalization and the Berkovich analytification

Functorial tropicalization of logarithmic schemes: the case of constant coefficients

Skeletons of stable maps II: superabundant geometries

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