On the structure of non-archimedean analytic curves

Baker, Matthew; Payne, Sam; Rabinoff, Joseph

doi:10.1090/conm/605/12113

Cited by 91 publications

(211 citation statements)

References 17 publications

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“…This proves (2). (2) shows that Z is d(ω)-dimensional and the same is true for in ω (X ) by Lemma 5.3.…”

Section: Annales De L'institut Fouriersupporting

confidence: 56%

“…Choose H). Moreover, the cell of S(X , H) corresponding to a cell (P, C) of HK(X, Y Σ ) maps to P under trop by Lemma 9.15, so we have proved (2).…”

Section: That (X H)mentioning

confidence: 70%

“…By [32, Théorème 5.3], every limit point of S Trop (X) is the limit of a sequence of points. (2) We use the notation from §3.1.…”

Section: Limit Points Of the Tropical Skeletonmentioning

confidence: 99%

“…If X is a curve, the problem of finding subgraphs of X an which are isometric to tropical varieties of X was investigated by Baker, Payne and Rabinoff in [2,3].…”

Section: Introductionmentioning

confidence: 99%

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Tropical Skeletons

Gubler¹,

Rabinoff²,

Werner³

2017

Ann. inst. Fourier

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In this paper, we study the interplay between tropical and analytic geometry for closed subschemes of toric varieties. Let K be a complete non-Archimedean field, and let X be a closed subscheme of a toric variety over K. We define the tropical skeleton of X as the subset of the associated Berkovich space X an which collects all Shilov boundary points in the fibers of the Kajiwara-Payne tropicalization map. We develop polyhedral criteria for limit points to belong to the tropical skeleton, and for the tropical skeleton to be closed. We apply the limit point criteria to the question of continuity of the canonical section of the tropicalization map on the multiplicity-one locus. This map is known to be continuous on all torus orbits; we prove criteria for continuity when crossing torus orbits. When X is schön and defined over a discretely valued field, we show that the tropical skeleton coincides with a skeleton of a strictly semistable pair, and is naturally isomorphic to the parameterizing complex of Helm-Katz.

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“…This proves (2). (2) shows that Z is d(ω)-dimensional and the same is true for in ω (X ) by Lemma 5.3.…”

Section: Annales De L'institut Fouriersupporting

confidence: 56%

“…Choose H). Moreover, the cell of S(X , H) corresponding to a cell (P, C) of HK(X, Y Σ ) maps to P under trop by Lemma 9.15, so we have proved (2).…”

Section: That (X H)mentioning

confidence: 70%

“…By [32, Théorème 5.3], every limit point of S Trop (X) is the limit of a sequence of points. (2) We use the notation from §3.1.…”

Section: Limit Points Of the Tropical Skeletonmentioning

confidence: 99%

“…If X is a curve, the problem of finding subgraphs of X an which are isometric to tropical varieties of X was investigated by Baker, Payne and Rabinoff in [2,3].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Tropical Skeletons

Gubler¹,

Rabinoff²,

Werner³

2017

Ann. inst. Fourier

Self Cite

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show abstract

“…This method to compute tropicalizations is due to [BPR13,BPR16] and we will refer to these papers for details of the following construction. Skeleta are discussed in [BPR13] and we refer to [BPR13, Corollary 4.23] for existence and uniqueness of the minimal skeleton S(W ) of the smooth curve W . We recall that the skeleton S(W ) has a canonical retraction τ : (P 1 K ) an → S(W ) and hence S(W ) is a compact subset of (P 1 K ) an .…”

Section: 17])mentioning

confidence: 99%

A tropical approach to nonarchimedean Arakelov geometry

Gubler

Kuennemann

2017

Alg. Number Th.

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Abstract. Chambert-Loir and Ducros have recently introduced a theory of real valued differential forms and currents on Berkovich spaces. In analogy to the theory of forms with logarithmic singularities, we enlarge the space of differential forms by so called δ-forms on the non-archimedean analytification of an algebraic variety. This extension is based on an intersection theory for tropical cycles with smooth weights. We prove a generalization of the Poincaré-Lelong formula which allows us to represent the first Chern current of a formally metrized line bundle by a δ-form. We introduce the associated Monge-Ampère measure µ as a wedge-power of this first Chern δ-form and we show that µ is equal to the corresponding Chambert-Loir measure. The * -product of Green currents is a crucial ingredient in the construction of the arithmetic intersection product. Using the formalism of δ-forms, we obtain a non-archimedean analogue at least in the case of divisors. We use it to compute non-archimedean local heights of proper varieties.

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Gonality of complete graphs with a small number of omitted edges

Panizzut

2016

Mathematische Nachrichten

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Let K d be the complete metric graph on d vertices. We compute the gonality of graphs obtained from K d by omitting edges forming a K h , or general configurations of at most d − 2 edges. We also investigate if these graphs can be lifted to curves with the same gonality. We lift the former graphs and the ones obtained by removing up to d − 2 edges not forming a K 3 using models of plane curves with certain singularities. We also study the gonality when removing d − 1 edges not forming a K 3 . We use harmonic morphism to lift these graphs to curves with the same gonality because in this case plane singular models can no longer be used due to a result of Coppens and Kato.

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On the structure of non-archimedean analytic curves

Cited by 91 publications

References 17 publications

Tropical Skeletons

Tropical Skeletons

A tropical approach to nonarchimedean Arakelov geometry

Gonality of complete graphs with a small number of omitted edges

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