The tropicalization  of the moduli space of curves

Abramovich, Dan; Caporaso, Lucia; Payne, Sam

doi:10.24033/asens.2258

The tropicalization of the moduli space of curves
[La tropicalisation de l'espace des modules des courbes]

Abramovich, Dan ; Caporaso, Lucia ; Payne, Sam

Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 765-809.

Résumé
Abstract

On démontre que le squelette du champ des modules des courbes stables de Deligne-Mumford-Knudsen est naturellement identifié avec l'espace des modules des courbes tropicales de façon compatible avec l'application de tropicalisation « naïve » d'ensembles. La démonstration emploie des résultats généraux de structure sur le squelette des champs toroïdaux de Deligne-Mumford. En outre, on construit les morphismes tautologiques entre les espaces de modules des courbes tropicales étendues, et on démontre qu'ils sont compatibles avec leurs analogues dans le cadre algébrique.

We show that the skeleton of the Deligne-Mumford-Knudsen moduli stack of stable curves is naturally identified with the moduli space of extended tropical curves, and that this is compatible with the “naive” set-theoretic tropicalization map. The proof passes through general structure results on the skeleton of a toroidal Deligne-Mumford stack. Furthermore, we construct tautological forgetful, clutching, and gluing maps between moduli spaces of extended tropical curves and show that they are compatible with the analogous tautological maps in the algebraic setting.

Publié le : 2015-09-24

MR Zbl

DOI : 10.24033/asens.2258

Classification : 14T05, 14H10, 14G22.
Keywords: Tropicalization, moduli of curves, Berkovich spaces, skeletons, toroidal Deligne-Mumford stacks.
Mot clés : Tropicalisation, espaces de modules des courbes, espaces de Berkovich, champs toroïdaux de Deligne-Mumford

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     author = {Abramovich, Dan and Caporaso, Lucia and Payne, Sam},
     title = {The tropicalization  of the moduli space of curves},
     journal = {Annales scientifiques de l'\'Ecole Normale Sup\'erieure},
     pages = {765--809},
     publisher = {Soci\'et\'e Math\'ematique de France. Tous droits r\'eserv\'es},
     volume = {Ser. 4, 48},
     number = {4},
     year = {2015},
     doi = {10.24033/asens.2258},
     mrnumber = {3377065},
     zbl = {1410.14049},
     language = {en},
     url = {http://www.numdam.org/articles/10.24033/asens.2258/}
}

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Abramovich, Dan; Caporaso, Lucia; Payne, Sam. The tropicalization  of the moduli space of curves. Annales scientifiques de l'École Normale Supérieure, Série 4, Tome 48 (2015) no. 4, pp. 765-809. doi : 10.24033/asens.2258. http://www.numdam.org/articles/10.24033/asens.2258/

Bibliographie
Cité par

Arbarello, E.; Cornalba, M.; Griffiths, P., Grund. Math. Wiss., 268, Springer, 2011, 963 pages (ISBN: 978-3-540-42688-2) | DOI | MR | Zbl

Abramovich, D.; Karu, K. Weak semistable reduction in characteristic 0, Invent. Math., Volume 139 (2000), pp. 241-273 (ISSN: 0020-9910) | DOI | MR | Zbl

Alexeev, V. Complete moduli in the presence of semiabelian group action, Ann. of Math., Volume 155 (2002), pp. 611-708 (ISSN: 0003-486X) | DOI | MR | Zbl

Abramovich, D.; Matsuki, K.; Rashid, S. A note on the factorization theorem of toric birational maps after Morelli and its toroidal extension, Tohoku Math. J., Volume 51 (1999), pp. 489-537 (ISSN: 0040-8735) | DOI | MR | Zbl

Borisov, L.; Chen, L.; Smith, G. The orbifold Chow ring of toric Deligne-Mumford stacks, J. Amer. Math. Soc., Volume 18 (2005), pp. 193-215 (ISSN: 0894-0347) | DOI | MR | Zbl

Berkovich, V., Mathematical Surveys and Monographs, 33, Amer. Math. Soc., 1990, 169 pages (ISBN: 0-8218-1534-2) | MR | Zbl

Berkovich, V. Smooth $p$ -adic analytic spaces are locally contractible, Invent. Math., Volume 137 (1999), pp. 1-84 (ISSN: 0020-9910) | DOI | MR | Zbl

Boucksom, S.; Favre, C.; Jonsson, M. Valuations and plurisubharmonic singularities, Publ. Res. Inst. Math. Sci., Volume 44 (2008), pp. 449-494 (ISSN: 0034-5318) | DOI | MR | Zbl

Behrend, K.; Manin, Y. I. Stacks of stable maps and Gromov-Witten invariants, Duke Math. J., Volume 85 (1996), pp. 1-60 (ISSN: 0012-7094) | DOI | MR | Zbl

Brannetti, S.; Melo, M.; Viviani, F. On the tropical Torelli map, Adv. Math., Volume 226 (2011), pp. 2546-2586 (ISSN: 0001-8708) | DOI | MR | Zbl

Baker, M.; Payne, S.; Rabinoff, J. Nonarchimedean geometry, tropicalization, and metrics on curves (preprint arXiv:1104.0320, to appear in J. Algebraic Geom ) | MR

Caporaso, L., Handbook of Moduli, Volume I (Advanced Lectures in Mathematics), Volume XXIV, International Press, Boston, 2012, pp. 119-160 | MR | Zbl

Chan, M. Combinatorics of the tropical Torelli map, Algebra Number Theory, Volume 6 (2012), pp. 1133-1169 (ISSN: 1937-0652) | DOI | MR | Zbl

Cavalieri, R.; Markwig, H.; Ranganathan, D. Tropicalizing the space of admissible covers (preprint arXiv:1401.4626 ) | MR

Chan, M.; Melo, M.; Viviani, F., Proceedings of the CIEM workshop in tropical geometry (Contemp. Math.), Volume 589, Amer. Math. Soc., 2013, pp. 45-85 | MR | Zbl

Conrad, B.; Temkin, M. Non-Archimedean analytification of algebraic spaces, J. Algebraic Geom., Volume 18 (2009), pp. 731-788 (ISSN: 1056-3911) | DOI | MR | Zbl

Caporaso, L.; Viviani, F. Torelli theorem for graphs and tropical curves, Duke Math. J., Volume 153 (2010), pp. 129-171 (ISSN: 0012-7094) | DOI | MR | Zbl

Culler, M.; Vogtmann, K. Moduli of graphs and automorphisms of free groups, Invent. Math., Volume 84 (1986), pp. 91-119 (ISSN: 0020-9910) | DOI | MR | Zbl

Denef, J. Some remarks on toroidal morphisms, 2012 https://perswww.kuleuven.be/~u0009256/Denef-Mathematics/denef_papers/ToroidalMorphisms.pdf ( https://perswww.kuleuven.be/~u0009256/Denef-Mathematics/denef_papers/ToroidalMorphisms.pdf )

Deligne, P.; Mumford, D. The irreducibility of the space of curves of given genus, Publ. Math. IHÉS, Volume 36 (1969), pp. 75-109 (ISSN: 0073-8301) | DOI | Numdam | MR | Zbl

Fulton, W., Annals of Math. Studies, 131, Princeton Univ. Press, 1993, 157 pages (The William H. Roever Lectures in Geometry) (ISBN: 0-691-00049-2) | MR | Zbl

Gathmann, A.; Kerber, M.; Markwig, H. Tropical fans and the moduli spaces of tropical curves, Compos. Math., Volume 145 (2009), pp. 173-195 (ISSN: 0010-437X) | DOI | MR | Zbl

Gathmann, A.; Markwig, H. Kontsevich's formula and the WDVV equations in tropical geometry, Adv. Math., Volume 217 (2008), pp. 537-560 (ISSN: 0001-8708) | DOI | MR | Zbl

Gross, M.; Siebert, B. Logarithmic Gromov-Witten invariants, J. Amer. Math. Soc., Volume 26 (2013), pp. 451-510 (ISSN: 0894-0347) | DOI | MR | Zbl

Hatcher, A., Cambridge Univ. Press, 2002, 544 pages (ISBN: 0-521-79160-X; 0-521-79540-0) | MR | Zbl

Hacking, P.; Keel, S.; Tevelev, J. Compactification of the moduli space of hyperplane arrangements, J. Algebraic Geom., Volume 15 (2006), pp. 657-680 (ISSN: 1056-3911) | DOI | MR | Zbl

Hacking, P.; Keel, S.; Tevelev, J. Stable pair, tropical, and log canonical compact moduli of Del Pezzo surfaces, Invent. Math., Volume 178 (2009), pp. 173-228 | DOI | MR | Zbl

Hrushovski, E.; Loeser, F. Non-archimedean tame topology and stably dominated types (preprint arXiv:1009.0252, to appear in Annals of Math. Studies ) | MR

Harris, J.; Morrison, I., Graduate Texts in Math., 187, Springer, 1998, 366 pages (ISBN: 0-387-98438-0; 0-387-98429-1) | MR | Zbl

Kajiwara, T., Toric topology (Contemp. Math.), Volume 460, Amer. Math. Soc., 2008, pp. 197-207 | DOI | MR | Zbl

Kato, K., Algebraic analysis, geometry, and number theory (Baltimore, MD, 1988), Johns Hopkins Univ. Press, 1989, pp. 191-224 | MR | Zbl

Kato, K. Toric singularities, Amer. J. Math., Volume 116 (1994), pp. 1073-1099 (ISSN: 0002-9327) | DOI | MR | Zbl

Kempf, G.; Knudsen, F.; Mumford, D.; Saint-Donat, B., Lecture Notes in Math., 339, Springer, 1973, 209 pages | MR | Zbl

Keel, S.; Mori, S. Quotients by groupoids, Ann. of Math., Volume 145 (1997), pp. 193-213 (ISSN: 0003-486X) | DOI | MR | Zbl

Kozlov, D. The topology of moduli spaces of tropical curves with marked points, Asian J. Math., Volume 13 (2009), pp. 385-403 (ISSN: 1093-6106) | DOI | MR | Zbl

Kozlov, D. Moduli spaces of tropical curves of higher genus with marked points and homotopy colimits, Israel J. Math., Volume 182 (2011), pp. 253-291 (ISSN: 0021-2172) | DOI | MR | Zbl

Lim, C. M.; Payne, S.; Potashnik, N. A note on Brill-Noether theory and rank-determining sets for metric graphs, Int. Math. Res. Not., Volume 2012 (2012), pp. 5484-5504 (ISSN: 1073-7928) | DOI | MR | Zbl

Mikhalkin, G., International Congress of Mathematicians. Vol. II, Eur. Math. Soc., Zürich, 2006, pp. 827-852 | MR | Zbl

Mikhalkin, G. Moduli spaces of rational tropical curves, Proceedings of Gökova Geometry-Topology Conference 2006, Gökova Geometry/Topology Conference (GGT), Gökova (2007), pp. 39-51 | MR | Zbl

Noohi, B. Fundamental groups of algebraic stacks, J. Inst. Math. Jussieu, Volume 3 (2004), pp. 69-103 (ISSN: 1474-7480) | DOI | MR | Zbl

Oda, T., Ergebn. Math. Grenzg., 15, Springer, 1988, 212 pages (ISBN: 3-540-17600-4) | MR | Zbl

Payne, S. Analytification is the limit of all tropicalizations, Math. Res. Lett., Volume 16 (2009), pp. 543-556 (ISSN: 1073-2780) | DOI | MR | Zbl

Payne, S. Toric vector bundles, branched covers of fans, and the resolution property, J. Algebraic Geom., Volume 18 (2009), pp. 1-36 (ISSN: 1056-3911) | MR | Zbl

Rabinoff, J. Tropical analytic geometry, Newton polygons, and tropical intersections, Adv. Math., Volume 229 (2012), pp. 3192-3255 (ISSN: 0001-8708) | DOI | MR | Zbl

Speyer, D.; Sturmfels, B. Tropical mathematics, Math. Mag., Volume 82 (2009), pp. 163-173 (ISSN: 0025-570X) | DOI | MR | Zbl

Tevelev, J. Compactifications of subvarieties of tori, Amer. J. Math., Volume 129 (2007), pp. 1087-1104 (ISSN: 0002-9327) | DOI | MR | Zbl

Thuillier, A. Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d'homotopie de certains schémas formels, Manuscripta Math., Volume 123 (2007), pp. 381-451 (ISSN: 0025-2611) | DOI | MR | Zbl

Tyomkin, I. Tropical geometry and correspondence theorems via toric stacks, Math. Ann., Volume 353 (2012), pp. 945-995 (ISSN: 0025-5831) | DOI | MR | Zbl

Ulirsch, M. Functorial tropicalization of logarithmic schemes: The case of constant coefficients (preprint arXiv:1310.6269 ) | MR

Viviani, F., Tropical and non-Archimedean geometry (Contemp. Math.), Volume 605, Amer. Math. Soc., Providence, RI, 2013, pp. 181-210 | DOI | MR | Zbl

Yu, T. Y. Gromov compactness in tropical geometry and in non-Archimedean analytic geometry (preprint arXiv:1401.6452 )

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