The universal tropicalization and the Berkovich analytification

Jeffrey Giansiracusa; Noah Giansiracusa

Kybernetika (2022)

  • Volume: 58, Issue: 5, page 790-815
  • ISSN: 0023-5954

Abstract

top
Given an integral scheme over a non-archimedean valued field , we construct a universal closed embedding of into a -scheme equipped with a model over the field with one element (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of by previous work of the authors, and we show that the set-theoretic tropicalization of with respect to this universal embedding is the Berkovich analytification . Moreover, using the scheme-theoretic tropicalization we previously introduced, we obtain a tropical scheme whose -points give the analytification and that canonically maps to all other scheme-theoretic tropicalizations of . This makes precise the idea that the Berkovich analytification is the universal tropicalization. When is affine, we show that is the limit of the tropicalizations of with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that represents the moduli functor of semivaluations on , and when is affine there is a universal semivaluation on taking values in the idempotent semiring of regular functions on the universal tropicalization.

How to cite

top

Giansiracusa, Jeffrey, and Giansiracusa, Noah. "The universal tropicalization and the Berkovich analytification." Kybernetika 58.5 (2022): 790-815. <http://eudml.org/doc/299458>.

@article{Giansiracusa2022,
abstract = {Given an integral scheme $X$ over a non-archimedean valued field $k$, we construct a universal closed embedding of $X$ into a $k$-scheme equipped with a model over the field with one element $\mathbb \{F\}_1$ (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of $X$ by previous work of the authors, and we show that the set-theoretic tropicalization of $X$ with respect to this universal embedding is the Berkovich analytification $X^\{\mathrm \{an\}\}$. Moreover, using the scheme-theoretic tropicalization we previously introduced, we obtain a tropical scheme $\mathit \{Trop\}_\{univ\}(X)$ whose $\mathbb \{T\}$-points give the analytification and that canonically maps to all other scheme-theoretic tropicalizations of $X$. This makes precise the idea that the Berkovich analytification is the universal tropicalization. When $X=\mathrm \{Spec\}\: A$ is affine, we show that $\mathit \{Trop\}_\{univ\}(X)$ is the limit of the tropicalizations of $X$ with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that $\mathit \{Trop\}_\{univ\}(X)$ represents the moduli functor of semivaluations on $X$, and when $X=\mathrm \{Spec\}\: A$ is affine there is a universal semivaluation on $A$ taking values in the idempotent semiring of regular functions on the universal tropicalization.},
author = {Giansiracusa, Jeffrey, Giansiracusa, Noah},
journal = {Kybernetika},
keywords = {tropical geometry; tropical schemes; idempotent semirings; Berkovich analytification; semivaluation},
language = {eng},
number = {5},
pages = {790-815},
publisher = {Institute of Information Theory and Automation AS CR},
title = {The universal tropicalization and the Berkovich analytification},
url = {http://eudml.org/doc/299458},
volume = {58},
year = {2022},
}

TY - JOUR
AU - Giansiracusa, Jeffrey
AU - Giansiracusa, Noah
TI - The universal tropicalization and the Berkovich analytification
JO - Kybernetika
PY - 2022
PB - Institute of Information Theory and Automation AS CR
VL - 58
IS - 5
SP - 790
EP - 815
AB - Given an integral scheme $X$ over a non-archimedean valued field $k$, we construct a universal closed embedding of $X$ into a $k$-scheme equipped with a model over the field with one element $\mathbb {F}_1$ (a generalization of a toric variety). An embedding into such an ambient space determines a tropicalization of $X$ by previous work of the authors, and we show that the set-theoretic tropicalization of $X$ with respect to this universal embedding is the Berkovich analytification $X^{\mathrm {an}}$. Moreover, using the scheme-theoretic tropicalization we previously introduced, we obtain a tropical scheme $\mathit {Trop}_{univ}(X)$ whose $\mathbb {T}$-points give the analytification and that canonically maps to all other scheme-theoretic tropicalizations of $X$. This makes precise the idea that the Berkovich analytification is the universal tropicalization. When $X=\mathrm {Spec}\: A$ is affine, we show that $\mathit {Trop}_{univ}(X)$ is the limit of the tropicalizations of $X$ with respect to all embeddings in affine space, thus giving a scheme-theoretic enrichment of a well-known result of Payne. Finally, we show that $\mathit {Trop}_{univ}(X)$ represents the moduli functor of semivaluations on $X$, and when $X=\mathrm {Spec}\: A$ is affine there is a universal semivaluation on $A$ taking values in the idempotent semiring of regular functions on the universal tropicalization.
LA - eng
KW - tropical geometry; tropical schemes; idempotent semirings; Berkovich analytification; semivaluation
UR - http://eudml.org/doc/299458
ER -

References

top
  1. Banerjee, S., , J. Reine Angew. Math. 2015 (2013), 71-87. DOI
  2. Berkovich, V. G., Spectral theory and analytic geometry over non-Archimedean fields., Mathematical Surveys and Monographs, American Mathematical Society 33, Providence 1990. 
  3. Berkovich, V. G., , Invent. Math. 137 (1999), 1, 1-84. DOI
  4. Berkovich, V. G., , In: Geometric aspects of Dwork theory. (Vol. I, II.), Walter de Gruyter, Berlin 2004, pp. 293-370. DOI
  5. Deitmar, A., Schemes over ., In: Number fields and function fields-two parallel worlds, Progr. Math. 239, Birkhäuser Boston, Boston, 2005, pp. 87-100. 
  6. Deitmar, A., -schemes and toric varieties., Beiträge Algebra Geom. 49 (2008), 2, 517-525. 
  7. Durov, N., New approach to arakelov geometry., arXiv:0704.2030, 2007. 
  8. Einsiedler, M., Kapranov, M., Lind, D., Non-Archimedean amoebas and tropical varieties., J. Reine Angew. Math. 601 (2006), 139-157. 
  9. Foster, T., Gross, P., Payne, S., , Israel J. Math. 201 (2014), 2, 835-846. DOI
  10. Foster, T., Ranganathan, D., , Manuscr. Math. 151 (2016), 3-4, 353-374. DOI
  11. Giansiracusa, J., Giansiracusa, N., 10.1215/00127094-3645544, Duke Math. J. 165 (2016), 18, 3379-3433. DOI10.1215/00127094-3645544
  12. Gubler, W., Rabinoff, J., Werner, A., Skeletons and tropicalizations., arXiv:1404.7044, 2014. 
  13. Hrushovski, E., Loeser, F., Non-archimedean tame topology and stably dominated types., Ann. Math. Studies 192, Princeton University Press, Princeton 2016. 
  14. Huber, R., Étale cohomology of rigid analytic varieties and adic spaces., Aspects of Mathematics E30, Friedr. Vieweg and Sohn, Braunschweig 1996. 
  15. Jun, J., Mincheva, K., Tolliver, J., Vector bundles on tropical schemes., arXiv:2009.03030, 2020. 
  16. Kajiwara, T., Tropical toric geometry., In: Contemp. Math., Toric topology 460, Amer. Math. Soc., Providence 2008, pp. 197-207. 
  17. Kato, K., 10.2307/2374941, Amer. J. Math. 116 (1994), 5, 1073-1099. DOI10.2307/2374941
  18. Kontsevich, M., Soibelman, Y., , In: Progr. Math., The Unity Math. 244, Birkhäuser Boston, Boston 2006. pp. 321-385. DOI
  19. Kontsevich, M., Tschinkel, Y., Non-archimedean kähler geometry. 
  20. Kuronya, A., Souza, P., Ulirsch, M., Tropicalization of toric prevarieties., arXiv:2107.03139, 2021. 
  21. Maclagan, D., Rincón, F., , Compos. Math. 154 (2018), 3, 640-670. DOI
  22. Maclagan, D., Rincón, F., , J. Eur. Math. Soc. (JEMS) 22 (2020), 3, 777-796. DOI
  23. Maclagan, D., Rincón, F., Varieties of tropical ideals are balanced., arXiv:2009.14557, 2020. 
  24. Maclagan, D., Sturmfels, B., Introduction to tropical geometry., American Mathematical Society, Graduate Studies in Mathematics 161, Providence 2015. 
  25. Macpherson, A. W., Skeleta in non-Archimedean and tropical geometry., arXiv:1311.0502, 2013. 
  26. Mikhalkin, G., Tropical Geometry., Unfinish draft book. 
  27. Mikhalkin, G., Tropical geometry and its applications., In: International Congress of Mathematicians II, Eur. Math. Soc., Zürich 2006, pp. 827-852. Zbl1103.14034
  28. Mustata, M., Nicaise, J., Weight functions on non-Archimedean analytic spaces and the Kontsevich-Soibelman skeleton., arXiv:1212.6328, 2013. 
  29. Payne, S., , Math. Res. Lett. 16 (2009), 3, 543-556. DOI
  30. Payne, S., , Math. Z. 262 (2009), 2, 301-311. DOI
  31. Popescu-Pampu, P., Stepanov, D., , In: Algebraic and combinatorial aspects of tropical geometry, Contemp. Math. 589, Amer. Math. Soc., Providence 2013, pp. 253-316. DOI
  32. Richter-Gebert, J., Sturmfels, B., Theobald, T., 10.1090/conm/377/06998, In: Idempotent mathematics and mathematical physics, Contemp. Math. 377, Amer. Math. Soc., Providence 2005, pp. 289-317. Zbl1093.14080DOI10.1090/conm/377/06998
  33. Temkin, M., , Israel J. Math. 185 (2011), 1-42. DOI
  34. Thuillier, A., Théorie du potentiel sur les courbes en géométrie analytique non archimédienne. applications à la théorie d'arakelov, 2005. 
  35. Toën, B., Vaquié, M., , J. K-Theory 3 (2009), 3, 437-500. DOI
  36. Ulirsch, M., , Proc. Lond. Math. Soc. 3 114 (2017), 6, 1081-1113. DOI
  37. Ulirsch, M., , Adv. Math. 345 (2019), 346-381. DOI
  38. Włodarczyk, J., Embeddings in toric varieties and prevarieties., J. Algebraic Geom. 2 (1993), 4, 705-726. 

NotesEmbed ?

top

You must be logged in to post comments.