Tropical Skeletons
Annales de l'Institut Fourier, Volume 67 (2017) no. 5, p. 1905-1961
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.
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.
Received : 2015-09-15
Revised : 2017-01-11
Accepted : 2017-01-24
Published online : 2017-11-17
DOI : https://doi.org/10.5802/aif.3125
Classification:  14G22,  14T05
Keywords: Tropical geometry, Kajiwara–Payne tropicalization, Berkovich spaces, skeletons
@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},
     publisher = {Association des Annales de l'institut Fourier},
     volume = {67},
     number = {5},
     year = {2017},
     pages = {1905-1961},
     doi = {10.5802/aif.3125},
     language = {en},
     url = {https://aif.centre-mersenne.org/item/AIF_2017__67_5_1905_0}
}
Tropical Skeletons. Annales de l'Institut Fourier, Volume 67 (2017) no. 5, pp. 1905-1961. doi : 10.5802/aif.3125. https://aif.centre-mersenne.org/item/AIF_2017__67_5_1905_0/

[1] Abramovich, Dan; Caporaso, Lucia; Payne, Sam The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér., Tome 48 (2015) no. 4, pp. 765-809 | Article | Zbl 06502666

[2] Baker, Matthew; Payne, Sam; Rabinoff, Joseph On the structure of non-Archimedean analytic curves, Tropical and Non-Archimedean Geometry, American Mathematical Society (Contemporary Mathematics) Tome 605 (2013), pp. 93-121 | Zbl 1320.14040

[3] Baker, Matthew; Payne, Sam; Rabinoff, Joseph Nonarchimedean geometry, tropicalization, and metrics on curves, Algebr. Geom., Tome 3 (2016) no. 1, pp. 63-105 | Article | Zbl 06609386

[4] Berkovich, Vladimir G. Spectral theory and analytic geometry over non-Archimedean fields, American Mathematical Society, Mathematical Surveys and Monographs, Tome 33 (1990), ix+169 pages | MR 1070709 (91k:32038) | Zbl 0715.14013

[5] Berkovich, Vladimir G. Étale cohomology for non-Archimedean analytic spaces, Publ. Math., Inst. Hautes Étud. Sci., Tome 78 (1993), pp. 5-161 | Article | MR 1259429 (95c:14017) | Zbl 0804.32019

[6] Berkovich, Vladimir G. Smooth p-adic analytic spaces are locally contractible, Invent. Math., Tome 137 (1999) no. 1, pp. 1-84 | Article | MR 1702143 (2000i:14028) | Zbl 0930.32016

[7] Berkovich, Vladimir G. An analog of Tate’s conjecture over local and finitely generated fields, Int. Math. Res. Not., Tome 2000 (2000) no. 13, pp. 665-680 | Article | MR 1772523 (2001h:14022) | Zbl 1068.14502

[8] Bieri, Robert; Groves, John R.J. The geometry of the set of characters induced by valuations, J. Reine Angew. Math., Tome 347 (1984), pp. 168-195 | MR 733052 (86c:14001) | Zbl 0526.13003

[9] Bosch, Siegfried; Güntzer, Ulrich; Remmert, Reinhold Non-Archimedean analysis, Springer, Grundlehren der Mathematischen Wissenschaften, Tome 261 (1984), xii+436 pages | Article | MR 746961 (86b:32031) | Zbl 0539.14017

[10] Bosch, Siegfried; Lütkebohmert, Werner Formal and rigid geometry. I. Rigid spaces, Math. Ann., Tome 295 (1993) no. 2, pp. 291-317 | Article | MR 1202394 (94a:11090) | Zbl 0808.14017

[11] Cavalieri, Renzo; Hampe, Simon; Markwig, Hannah; Ranganathan, Dhruv Moduli spaces of rational weighted stable curves and tropical geometry, Forum Math. Sigma, Tome 4 (2016) (ID e9, 35 pp.) | Article | Zbl 06601300

[12] Conrad, Brian Irreducible components of rigid spaces, Ann. Inst. Fourier, Tome 49 (1999) no. 2, pp. 473-541 | Article | MR 1697371 (2001c:14045) | Zbl 0928.32011

[13] Cox, David A.; Little, John B.; Schenck, Henry K. Toric varieties, American Mathematical Society, Graduate Studies in Mathematics, Tome 124 (2011), xxiv+841 pages | MR 2810322 (2012g:14094) | Zbl 1223.14001

[14] Cueto, Maria Angelica; Häbich, Mathias; Werner, Annette Faithful tropicalization of the Grassmannian of planes, Math. Ann., Tome 360 (2014) no. 1-2, pp. 391-437 | Article | MR 3263167 | Zbl 1310.14049

[15] Draisma, Jan; Postinghel, Elisa Faithful tropicalisation and torus actions, Manuscr. Math., Tome 149 (2016) no. 3-4, pp. 315-338 | Article | MR 3458171 | Zbl 1342.14128

[16] Ducros, Antoine Image réciproque du squelette par un morphisme entre espaces de Berkovich de même dimension, Bull. Soc. Math. Fr., Tome 131 (2003) no. 4, pp. 483-506 | Article | MR 2044492 (2004m:14042) | Zbl 1068.14024

[17] Ducros, Antoine Les espaces de Berkovich sont excellents, Ann. Inst. Fourier, Tome 59 (2009) no. 4, pp. 1443-1552 http://aif.cedram.org/item?id=AIF_2009__59_4_1443_0 | Article | MR 2566967 (2011a:14048) | Zbl 1177.14049

[18] Ducros, Antoine Espaces de Berkovich, polytopes, squelettes et théorie des modèles, Confluentes Math., Tome 4 (2012) no. 4 (Paper 1250007, 57 pp., erratum ibid. 5 (2013), no. 2, p. 43-44) | Article | MR 3020334 | Zbl 1263.14030

[19] Foster, Tyler; Gross, Philipp; Payne, Sam Limits of tropicalizations, Isr. J. Math., Tome 201 (2014) no. 2, pp. 835-846 | Article | MR 3265305 | Zbl 1319.14059

[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 0199181 (33 #7330) | Zbl 0135.39701

[21] Gubler, Walter Tropical varieties for non-Archimedean analytic spaces, Invent. Math., Tome 169 (2007) no. 2, pp. 321-376 | Article | MR 2318559 (2008k:14085) | Zbl 1153.14036

[22] Gubler, Walter A guide to tropicalizations, Algebraic and combinatorial aspects of tropical geometry, American Mathematical Society (Contemporary Mathematics) Tome 589 (2013), pp. 125-189 | Article | MR 3088913 | Zbl 1318.14061

[23] Gubler, Walter; Rabinoff, Joseph; Werner, Annette Skeletons and tropicalizations, Adv. Math., Tome 294 (2016), pp. 150-215 | Article | Zbl 06567870

[24] Helm, David; Katz, Eric Monodromy filtrations and the topology of tropical varieties, Can. J. Math., Tome 64 (2012) no. 4, pp. 845-868 | Article | MR 2957233 | Zbl 1312.14145

[25] De Jong, Aise Johan Smoothness, semi-stability and alterations, Publ. Math., Inst. Hautes Étud. Sci. (1996) no. 83, pp. 51-93 | Article | MR 1423020 (98e:14011) | Zbl 0916.14005

[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., Tome 165 (2016) no. 16, pp. 3189-3240 | Article | Zbl 06666955

[27] Luxton, Mark; Qu, Zhenhua Some results on tropical compactifications, Trans. Am. Math. Soc., Tome 363 (2011) no. 9, pp. 4853-4876 | Article | MR 2806694 (2012g:14096) | Zbl 1230.14014

[28] Osserman, Brian; Payne, Sam Lifting tropical intersections, Doc. Math., J. DMV, Tome 18 (2013), pp. 121-175 | MR 3064984 | Zbl 1308.14069

[29] Osserman, Brian; Rabinoff, Joseph Lifting nonproper tropical intersections, Tropical and Non-Archimedean Geometry, American Mathematical Society, Providence, RI (Contemporary Mathematics) Tome 605 (2013), pp. 15-44 | Zbl 1320.14078

[30] Palais, Richard S. When proper maps are closed, Proc. Am. Math. Soc., Tome 24 (1970), p. 835-836 | MR 0254818 (40 #8025) | Zbl 0189.53202

[31] Payne, Sam Analytification is the limit of all tropicalizations, Math. Res. Lett., Tome 16 (2009) no. 2-3, pp. 543-556 | Article | MR 2511632 (2010j:14104) | Zbl 1193.14077

[32] Poineau, Jérôme Les espaces de Berkovich sont angéliques, Bull. Soc. Math. Fr., Tome 141 (2013) no. 2, pp. 267-297 | Article | Zbl 1314.14046

[33] Rabinoff, Joseph Tropical analytic geometry, Newton polygons, and tropical intersections, Adv. Math., Tome 229 (2012) no. 6, pp. 3192-3255 | Article | MR 2900439 | Zbl 1285.14072

[34] Speyer, David; Sturmfels, Bernd The tropical Grassmannian, Adv. Geom., Tome 4 (2004) no. 3, pp. 389-411 | Article | MR 2071813 (2005d:14089) | Zbl 1065.14071

[35] Temkin, Michael On local properties of non-Archimedean analytic spaces. II, Isr. J. Math., Tome 140 (2004), pp. 1-27 | Article | MR 2054837 (2005c:14030) | Zbl 1066.32025

[36] Tevelev, Jenia Compactifications of subvarieties of tori, Am. J. Math., Tome 129 (2007) no. 4, pp. 1087-1104 | Article | MR 2343384 (2008f:14068) | Zbl 1154.14039

[37] Werner, Annette Analytification and tropicalization over non-Archimedean fields, Nonarchimedean and Tropical Geometry, Springer (2016), pp. 123-174 | Zbl 1349.14103