Faithful tropicalization of the Grassmannian of planes

Cueto, María Angélica; Häbich, Mathias; Werner, Annette

doi:10.1007/s00208-014-1037-3

Cited by 17 publications

(40 citation statements)

References 29 publications

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“…With the help of the results in [14, §4 and 5], one can show that the projection of C T to the cocharacter space of the torus orbit E J is d-maximal. Moreover, it is shown in [14,Corollary 6.5] that the tropical Grassmannian has tropical multiplicity one everywhere. Therefore we can apply Theorem 8.14 to deduce the existence of a continous section to the tropicalization map.…”

Section: A Sequential Continuity Criterionmentioning

confidence: 99%

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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Section: A Sequential Continuity Criterionmentioning

confidence: 99%

Tropical Skeletons

Gubler¹,

Rabinoff²,

Werner³

2017

Ann. inst. Fourier

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“…A version of this theorem first appeared in [7]; see Remark 5.3 below. Lifts from Trop( Gr(2, m)) into tropicalisations of other flag varieties were constructed in [15,16].…”

Section: Theorem 51 the Surjective Projection From Z ⊆ Gr(2 M) An mentioning

confidence: 99%

“…As, by assumption, the minimal-weight edge in remains the edge i 0 j 0 , the minimal η-weight of an edge of with both vertices in J must be at least Remark 5.3. In [7] the setting is projective rather than affine. Theorem 1.1 and Corollary 7.3 from that paper follow from our theorem by applying Lemmas 3.5 and 3.4, respectively.…”

Section: We Now Argue That For Eachmentioning

confidence: 99%

“…In this paper we discuss a number of high-dimensional examples where trop has a continuous section. The results are motivated by exciting recent work for Grassmannians of planes [7] and for curves [5]. In particular, we will give another, more geometric proof of the main result of [7] that Grassmannians of planes admit such a section.…”

Section: Introductionmentioning

confidence: 99%

“…The results are motivated by exciting recent work for Grassmannians of planes [7] and for curves [5]. In particular, we will give another, more geometric proof of the main result of [7] that Grassmannians of planes admit such a section. In the recent paper [13] (written concurrently with our paper) it is proved that, if X is a subvariety of G n m , then a section exists on the locus in Trop(X ) where the tropical multiplicity equals one [13].…”

Section: Introductionmentioning

confidence: 99%

See 2 more Smart Citations

Faithful tropicalisation and torus actions

Draisma

Postinghel

2015

manuscripta math.

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Abstract.For any affine variety equipped with coordinates, there is a surjective, continuous map from its Berkovich space to its tropicalisation. Exploiting torus actions, we develop techniques for finding an explicit, continuous section of this map. In particular, we prove that such a section exists for linear spaces, Grassmannians of planes (reproving a result due to Cueto, Häbich, and Werner), matrix varieties defined by the vanishing of 3 × 3-minors, and for the hypersurface defined by Cayley's hyperdeterminant.

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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 tropicalization of the Grassmannian of planes

Cited by 17 publications

References 29 publications

Tropical Skeletons

Tropical Skeletons

Faithful tropicalisation and torus actions

Analytification and Tropicalization Over Non-archimedean Fields

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