Faithful tropicalisation and torus actions

Draisma, Jan; Postinghel, Elisa

doi:10.1007/s00229-015-0781-3

Cited by 9 publications

(6 citation statements)

References 22 publications

(37 reference statements)

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“…Note that Theorem 10.6 does not imply the continuity of the section on the whole tropical Grassmannian T Gr(2, n). Draisma and Postinghel [DP16] reprove this result with different techniques and use torus actions to obtain sections of the tropicalization map in other explicit situations.…”

Section: Sturmfels-tevelev Multiplicity Formulamentioning

confidence: 99%

Skeletons and tropicalizations

Gubler

Rabinoff

Werner

2016

Advances in Mathematics

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ABSTRACT. Let K be a complete, algebraically closed non-archimedean field with ring of integers K • and let X be a K-variety. We associate to the data of a strictly semistable K • -model X of X plus a suitable horizontal divisor H a skeleton S(X , H) in the analytification of X. This generalizes Berkovich's original construction by admitting unbounded faces in the directions of the components of H. It also generalizes constructions by Tyomkin and Baker-Payne-Rabinoff from curves to higher dimensions. Every such skeleton has an integral polyhedral structure. We show that the valuation of a non-zero rational function is piecewise linear on S(X , H). For such functions we define slopes along codimension one faces and prove a slope formula expressing a balancing condition on the skeleton. Moreover, we obtain a multiplicity formula for skeletons and tropicalizations in the spirit of a wellknown result by Sturmfels-Tevelev. We show a faithful tropicalization result saying roughly that every skeleton can be seen in a suitable tropicalization. We also prove a general result about existence and uniqueness of a continuous section to the tropicalization map on the locus of tropical multiplicity one.

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Section: Sturmfels-tevelev Multiplicity Formulamentioning

confidence: 99%

Skeletons and tropicalizations

Gubler

Rabinoff

Werner

2016

Advances in Mathematics

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“…For further details and examples, we refer the reader to the expanded version of this paper at arXiv:1104.0320. Since this paper was written, there have been a large number of follow-up articles: the reader may also want to read [GRW13], in which many of the results in this article are extended to higher dimensions; [CHW14,DP14,CS13], in which several interesting examples of faithful tropicalizations are given; and [ABBR14a, ABBR14b], in which "relative" versions of some of the results in this paper are used to prove tropical lifting theorems.…”

Section: Introductionmentioning

confidence: 99%

Nonarchimedean geometry, tropicalization, and metrics on curves

Baker¹,

Payne²,

Rabinoff³

2016

Alg. Geom.

105

227

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We develop a number of general techniques for comparing analytifications and tropicalizations of algebraic varieties. Our basic results include a projection formula for tropical multiplicities and a generalization of the Sturmfels-Tevelev multiplicity formula in tropical elimination theory to the case of a nontrivial valuation. For curves, we explore in detail the relationship between skeletal metrics and lattice lengths on tropicalizations and show that the maps from the analytification of a curve to the tropicalizations of its toric embeddings stabilize to isometries on finite subgraphs. Other applications include generalizations of Speyer's well-spacedness condition and the KatzMarkwig-Markwig results on tropical j-invariants.

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“…In [DP14], Draisma and Postinghel give a different proof of Theorem 4.1. They work with the affine cone over the Grassmannian Gr(2, n), define the section on a suitable subset and use torus actions and tropicalized torus actions to move it around.…”

Section: 2mentioning

confidence: 99%

Analytification and Tropicalization Over Non-archimedean Fields

Werner

2016

Nonarchimedean and Tropical Geometry

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Abstract:In this paper, we provide an overview of recent progress on the interplay between tropical geometry and non-archimedean analytic geometry in the sense of Berkovich. After briefly discussing results by Baker, Payne and Rabinoff [BPR11] [BPR13] in the case of curves, we explain a result from [CHW14] comparing the tropical Grassmannian of planes to the analytic Grassmannian. We also give an overview of most of the results in [GRW14], where a general higher-dimensional theory is developed. In particular, we explain the construction of generalized skeleta in [GRW14] which are polyhedral substructures of Berkovich spaces lending themselves to comparison with tropicalizations. We discuss the slope formula for the valuation of rational functions and explain two results on the comparison between polyhedral substructures of Berkovich spaces and tropicalizations.

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Faithful tropicalisation and torus actions

Cited by 9 publications

References 22 publications

Skeletons and tropicalizations

Skeletons and tropicalizations

Nonarchimedean geometry, tropicalization, and metrics on curves

Analytification and Tropicalization Over Non-archimedean Fields

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