Tropical Skeletons

Gubler, Walter; Rabinoff, Joseph; Werner, Annette

doi:10.5802/aif.3125

Tropical Skeletons
[Squelettes tropicaux]

Gubler, Walter ¹ ; Rabinoff, Joseph ² ; Werner, Annette ³

¹ Fakultät für Mathematik Universität Regensburg Universitätsstraße 31 D-93040 Regensburg (Germany)
² School of Mathematics Georgia Institute of Technology Atlanta GA 30332-0160 (USA)
³ Institut für Mathematik Goethe-Universität Frankfurt Robert-Mayer-Straße 8 D-60325 Frankfurt a.M. (Germany)

Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 1905-1961.

Résumé
Abstract

Nous étudions les relations entre la géométrie tropicale et la géométrie analytique pour les sous-schémas fermés des variétés toriques. Soit $K$ un corps non-archimédien et complet et soit $X$ un sous-schéma fermé d’une variété torique sur $K$ . Nous définissons le squelette tropical de X comme le sous-ensemble de l’espace de Berkovich associé $X^{an}$ qui est composé de tous les points du bord de Shilov dans les fibres du morphisme de tropicalisation de Kajiwara–Payne. Nous développons des critères polyèdraux pour que des points limite appartiennent au squelette tropical, et pour que cet espace soit fermé. Nous appliquons ce critère pour les points limite à la question de la continuité de la section canonique du morphisme de tropicalisation sur le lieu de multiplicité un. On sait que cette section est continue sur chaque orbite du tore ; nous donnons des critères de continuité au croisement des orbites. Quand $X$ est schön et défini sur un corps discrètement valué, nous montrons que la squelette tropical coïncide avec le squelette d’une paire strictement semistable, et qu’il est naturellement isomorphe au complexe paramétrisant de Helm–Katz.

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.

Reçu le : 2015-09-15
Révisé le : 2017-01-11
Accepté le : 2017-01-24
Publié le : 2017-11-17

DOI : 10.5802/aif.3125

Classification : 14G22, 14T05
Keywords: Tropical geometry, Kajiwara–Payne tropicalization, Berkovich spaces, skeletons
Mot clés : géométrie tropicale, tropicalisation de Kajiwara–Payne, espaces de Berkovich, squelettes

Affiliations des auteurs :

Gubler, Walter ¹ ; Rabinoff, Joseph ² ; Werner, Annette ³

Licence :

CC-BY-ND 4.0

Droits d'auteur : Les auteurs conservent leurs droits

@article{AIF_2017__67_5_1905_0,
     author = {Gubler, Walter and Rabinoff, Joseph and Werner, Annette},
     title = {Tropical {Skeletons}},
     journal = {Annales de l'Institut Fourier},
     pages = {1905--1961},
     publisher = {Association des Annales de l{\textquoteright}institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     doi = {10.5802/aif.3125},
     language = {en},
     url = {https://aif.centre-mersenne.org/articles/10.5802/aif.3125/}
}

TY  - JOUR
AU  - Gubler, Walter
AU  - Rabinoff, Joseph
AU  - Werner, Annette
TI  - Tropical Skeletons
JO  - Annales de l'Institut Fourier
PY  - 2017
SP  - 1905
EP  - 1961
VL  - 67
IS  - 5
PB  - Association des Annales de l’institut Fourier
UR  - https://aif.centre-mersenne.org/articles/10.5802/aif.3125/
DO  - 10.5802/aif.3125
LA  - en
ID  - AIF_2017__67_5_1905_0
ER  -

%0 Journal Article
%A Gubler, Walter
%A Rabinoff, Joseph
%A Werner, Annette
%T Tropical Skeletons
%J Annales de l'Institut Fourier
%D 2017
%P 1905-1961
%V 67
%N 5
%I Association des Annales de l’institut Fourier
%U https://aif.centre-mersenne.org/articles/10.5802/aif.3125/
%R 10.5802/aif.3125
%G en
%F AIF_2017__67_5_1905_0

Gubler, Walter; Rabinoff, Joseph; Werner, Annette. Tropical Skeletons. Annales de l'Institut Fourier, Tome 67 (2017) no. 5, pp. 1905-1961. doi : 10.5802/aif.3125. https://aif.centre-mersenne.org/articles/10.5802/aif.3125/

Bibliographie
Cité par

[1] Abramovich, Dan; Caporaso, Lucia; Payne, Sam The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér., Volume 48 (2015) no. 4, pp. 765-809 | DOI | Zbl

[2] Baker, Matthew; Payne, Sam; Rabinoff, Joseph On the structure of non-Archimedean analytic curves, Tropical and Non-Archimedean Geometry (Contemporary Mathematics), Volume 605, American Mathematical Society, 2013, pp. 93-121 | Zbl

[3] Baker, Matthew; Payne, Sam; Rabinoff, Joseph Nonarchimedean geometry, tropicalization, and metrics on curves, Algebr. Geom., Volume 3 (2016) no. 1, pp. 63-105 | DOI | Zbl

[4] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, 33, American Mathematical Society, 1990, ix+169 pages | MR | Zbl

[5] Berkovich, Vladimir G. Étale cohomology for non-Archimedean analytic spaces, Publ. Math., Inst. Hautes Étud. Sci., Volume 78 (1993), pp. 5-161 | DOI | MR | Zbl

[6] Berkovich, Vladimir G. Smooth $p$ -adic analytic spaces are locally contractible, Invent. Math., Volume 137 (1999) no. 1, pp. 1-84 | DOI | MR | Zbl

[7] Berkovich, Vladimir G. An analog of Tate’s conjecture over local and finitely generated fields, Int. Math. Res. Not., Volume 2000 (2000) no. 13, pp. 665-680 | DOI | MR | Zbl

[8] Bieri, Robert; Groves, John R.J. The geometry of the set of characters induced by valuations, J. Reine Angew. Math., Volume 347 (1984), pp. 168-195 | MR | Zbl

[9] Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold Non-Archimedean analysis, Grundlehren der Mathematischen Wissenschaften, 261, Springer, 1984, xii+436 pages | DOI | MR | Zbl

[10] Bosch, Siegfried; Lütkebohmert, Werner Formal and rigid geometry. I. Rigid spaces, Math. Ann., Volume 295 (1993) no. 2, pp. 291-317 | DOI | MR | Zbl

[11] Cavalieri, Renzo; Hampe, Simon; Markwig, Hannah; Ranganathan, Dhruv Moduli spaces of rational weighted stable curves and tropical geometry, Forum Math. Sigma, Volume 4 (2016) (ID e9, 35 pp.) | DOI | Zbl

[12] Conrad, Brian Irreducible components of rigid spaces, Ann. Inst. Fourier, Volume 49 (1999) no. 2, pp. 473-541 | DOI | MR | Zbl

[13] Cox, David A.; Little, John B.; Schenck, Henry K. Toric varieties, Graduate Studies in Mathematics, 124, American Mathematical Society, 2011, xxiv+841 pages | MR | Zbl

[14] Cueto, Maria Angelica; Häbich, Mathias; Werner, Annette Faithful tropicalization of the Grassmannian of planes, Math. Ann., Volume 360 (2014) no. 1-2, pp. 391-437 | DOI | MR | Zbl

[15] Draisma, Jan; Postinghel, Elisa Faithful tropicalisation and torus actions, Manuscr. Math., Volume 149 (2016) no. 3-4, pp. 315-338 | DOI | MR | Zbl

[16] Ducros, Antoine Image réciproque du squelette par un morphisme entre espaces de Berkovich de même dimension, Bull. Soc. Math. Fr., Volume 131 (2003) no. 4, pp. 483-506 | DOI | MR | Zbl

[17] Ducros, Antoine Les espaces de Berkovich sont excellents, Ann. Inst. Fourier, Volume 59 (2009) no. 4, pp. 1443-1552 http://aif.cedram.org/item?id=AIF_2009__59_4_1443_0 | DOI | MR | Zbl

[18] Ducros, Antoine Espaces de Berkovich, polytopes, squelettes et théorie des modèles, Confluentes Math., Volume 4 (2012) no. 4 Paper 1250007, 57 pp., erratum ibid. 5 (2013), no. 2, p. 43-44 | DOI | MR | Zbl

[19] Foster, Tyler; Gross, Philipp; Payne, Sam Limits of tropicalizations, Isr. J. Math., Volume 201 (2014) no. 2, pp. 835-846 | DOI | MR | Zbl

[20] Grothendieck, Alexander Éléments de géométrie algébrique. IV. Étude locale des schémas et des morphismes de schémas. II, Publ. Math., Inst. Hautes Étud. Sci. (1965) no. 24, pp. 1-231 | MR | Zbl

[21] Gubler, Walter Tropical varieties for non-Archimedean analytic spaces, Invent. Math., Volume 169 (2007) no. 2, pp. 321-376 | DOI | MR | Zbl

[22] Gubler, Walter A guide to tropicalizations, Algebraic and combinatorial aspects of tropical geometry (Contemporary Mathematics), Volume 589, American Mathematical Society, 2013, pp. 125-189 | DOI | MR | Zbl

[23] Gubler, Walter; Rabinoff, Joseph; Werner, Annette Skeletons and tropicalizations, Adv. Math., Volume 294 (2016), pp. 150-215 | DOI | Zbl

[24] Helm, David; Katz, Eric Monodromy filtrations and the topology of tropical varieties, Can. J. Math., Volume 64 (2012) no. 4, pp. 845-868 | DOI | MR | Zbl

[25] de Jong, Aise Johan Smoothness, semi-stability and alterations, Publ. Math., Inst. Hautes Étud. Sci. (1996) no. 83, pp. 51-93 | DOI | MR | Zbl

[26] Katz, Eric; Rabinoff, Joseph; Zureick-Brown, David Uniform bounds for the number of rational points on curves of small Mordell–Weil rank, Duke Math. J., Volume 165 (2016) no. 16, pp. 3189-3240 | DOI | Zbl

[27] Luxton, Mark; Qu, Zhenhua Some results on tropical compactifications, Trans. Am. Math. Soc., Volume 363 (2011) no. 9, pp. 4853-4876 | DOI | MR | Zbl

[28] Osserman, Brian; Payne, Sam Lifting tropical intersections, Doc. Math., J. DMV, Volume 18 (2013), pp. 121-175 | MR | Zbl

[29] Osserman, Brian; Rabinoff, Joseph Lifting nonproper tropical intersections, Tropical and Non-Archimedean Geometry (Contemporary Mathematics), Volume 605, American Mathematical Society, Providence, RI, 2013, pp. 15-44 | Zbl

[30] Palais, Richard S. When proper maps are closed, Proc. Am. Math. Soc., Volume 24 (1970), pp. 835-836 | MR | Zbl

[31] Payne, Sam Analytification is the limit of all tropicalizations, Math. Res. Lett., Volume 16 (2009) no. 2-3, pp. 543-556 | DOI | MR | Zbl

[32] Poineau, Jérôme Les espaces de Berkovich sont angéliques, Bull. Soc. Math. Fr., Volume 141 (2013) no. 2, pp. 267-297 | DOI | Zbl

[33] Rabinoff, Joseph Tropical analytic geometry, Newton polygons, and tropical intersections, Adv. Math., Volume 229 (2012) no. 6, pp. 3192-3255 | DOI | MR | Zbl

[34] Speyer, David; Sturmfels, Bernd The tropical Grassmannian, Adv. Geom., Volume 4 (2004) no. 3, pp. 389-411 | DOI | MR | Zbl

[35] Temkin, Michael On local properties of non-Archimedean analytic spaces. II, Isr. J. Math., Volume 140 (2004), pp. 1-27 | DOI | MR | Zbl

[36] Tevelev, Jenia Compactifications of subvarieties of tori, Am. J. Math., Volume 129 (2007) no. 4, pp. 1087-1104 | DOI | MR | Zbl

[37] Werner, Annette Analytification and tropicalization over non-Archimedean fields, Nonarchimedean and Tropical Geometry, Springer, 2016, pp. 123-174 | Zbl

Cité par Sources :

ANNALES DE L'INSTITUT FOURIER

ARTICLES À PARAÎTRE

FEUILLETER

PRESENTATION

COMITÉ DE RÉDACTION

SOUMETTRE UN ARTICLE

ABONNEMENT