Hostname: page-component-586b7cd67f-g8jcs Total loading time: 0 Render date: 2024-11-22T00:06:59.812Z Has data issue: false hasContentIssue false
  • English
  • Français

A MODULI STACK OF TROPICAL CURVES

Part of: Algebraic geometry: Foundations

Published online by Cambridge University Press:  24 April 2020

RENZO CAVALIERI ,
MELODY CHAN ,
MARTIN ULIRSCH  and
JONATHAN WISE
RENZO CAVALIERI
Affiliation:
Department of Mathematics, Colorado State University, Fort Collins, Colorado80523-1874, USA; [email protected]
MELODY CHAN
Affiliation:
Department of Mathematics, Brown University, Providence, Rhode Island02912, USA; [email protected]
MARTIN ULIRSCH
Affiliation:
Institut für Mathematik, Goethe-Universität Frankfurt, 60325Frankfurt am Main, Germany; [email protected]
JONATHAN WISE
Affiliation:
Department of Mathematics, University of Colorado, Boulder, Boulder,Colorado80309-0395, USA; [email protected]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

We contribute to the foundations of tropical geometry with a view toward formulating tropical moduli problems, and with the moduli space of curves as our main example. We propose a moduli functor for the moduli space of curves and show that it is representable by a geometric stack over the category of rational polyhedral cones. In this framework, the natural forgetful morphisms between moduli spaces of curves with marked points function as universal curves.

Our approach to tropical geometry permits tropical moduli problems—moduli of curves or otherwise—to be extended to logarithmic schemes. We use this to construct a smooth tropicalization morphism from the moduli space of algebraic curves to the moduli space of tropical curves, and we show that this morphism commutes with all of the tautological morphisms.

MSC classification

Primary: 14T05: Tropical geometry
Secondary: 14A20: Generalizations (algebraic spaces, stacks)
Type
Algebraic and Complex Geometry
Information
Forum of Mathematics, Sigma , Volume 8 , 2020 , e23
Creative Commons
Creative Common License - CCCreative Common License - BY
This is an Open Access article, distributed under the terms of the Creative Commons Attribution licence (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted re-use, distribution, and reproduction in any medium, provided the original work is properly cited.
Copyright
© The Author(s) 2020

References

Amini, O. and Baker, M., ‘Linear series on metrized complexes of algebraic curves’, Math. Ann. 362(1–2) (2015), 55–106.CrossRefGoogle Scholar
Abramovich, D., Chen, Q., Gillam, D., Huang, Y., Olsson, M., Satriano, M. and Sun, S., ‘Logarithmic geometry and moduli’, inHandbook of Moduli, Vol. I, Adv. Lect. Math. (ALM) 24 (Int. Press, Somerville, MA, 2013), 1–61.Google Scholar
Abramovich, D., Chen, Q., Marcus, S., Ulirsch, M. and Wise, J., ‘Skeletons and fans of logarithmic structures’, inNon-Archimedean and Tropical Geometry (eds. Baker, M. and Payne, S.) (Simons Symposia, Springer, 2016), 287–336.CrossRefGoogle Scholar
Abramovich, D., Chen, Q., Marcus, S. and Wise, J., ‘Boundedness of the space of stable logarithmic maps’, J. Eur. Math. Soc. (JEMS) 19(9) (2017), 2783–2809.CrossRefGoogle Scholar
Abramovich, D., Caporaso, L. and Payne, S., ‘The tropicalization of the moduli space of curves’, Ann. Sci. Éc. Norm. Supér. (4) 48(4) (2015), 765–809.CrossRefGoogle Scholar
Abramovich, D., Corti, A. and Vistoli, A., ‘Twisted bundles and admissible covers’, Comm. Algebra 31(8) (2003), 3547–3618. Special issue in honor of Steven L. Kleiman.CrossRefGoogle Scholar
Abramovich, D. and Karu, K., ‘Weak semistable reduction in characteristic 0’, Invent. Math. 139(2) (2000), 241–273.CrossRefGoogle Scholar
Abramovich, D., Marcus, S. and Wise, J., ‘Comparison theorems for Gromov-Witten invariants of smooth pairs and of degenerations’, Ann. Inst. Fourier (Grenoble) 64(4) (2014), 1611–1667.CrossRefGoogle Scholar
Abramovich, D. and Wise, J., ‘Birational invariance in logarithmic Gromov-Witten theory’, Compos. Math. 154(3) (2018), 595–620.CrossRefGoogle Scholar
Berkovich, V. G., Spectral theory and analytic geometry over non-Archimedean fields, Mathematical Surveys and Monographs, vol. 33. (American Mathematical Society, Providence, RI, 1990).Google Scholar
Berkovich, V. G., ‘Étale cohomology for non-Archimedean analytic spaces’, Inst. Hautes Études Sci. Publ. Math. 78 (1993), 5–161. (1994).CrossRefGoogle Scholar
Behrend, K., Ginot, G., Noohi, B. and Xu, P., ‘String topology for stacks’, Astérisque (343) (2012), xiv+169.Google Scholar
Brannetti, S., Melo, M. and Viviani, F., ‘On the tropical Torelli map’, Adv. Math. 226(3) (2011), 2546–2586.CrossRefGoogle Scholar
Caporaso, L., ‘Algebraic and tropical curves: comparing their moduli spaces’, inHandbook of Moduli. Vol. I, Adv. Lect. Math. (ALM) 24 (Int. Press, Somerville, MA, 2013), 119–160.Google Scholar
Caporaso, L., Algebraic geometry: Salt Lake City 2015, Proc. Sympos. Pure Math., 97.2 (Amer. Math. Soc., Providence, RI, 2018), 103–138.CrossRefGoogle Scholar
Chan, M., Faber, C., Galatius, S. and Payne, S., ‘The $S_{n}$-equivariant top weight Euler characteristic of $M_{g,n}$’, Preprint, 2019, arXiv:1904.06367 [math].Google Scholar
Chan, M., Galatius, S. and Payne, S., ‘Tropical curves, graph complexes, and top weight cohomology of $M_{g}$’, Preprint, 2018, arXiv:1805.10186 [math].Google Scholar
Chan, M., Galatius, S. and Payne, S., ‘Topology of moduli spaces of tropical curves with marked points’, Preprint, 2019, arXiv:1903.07187 [math].Google Scholar
Chan, M., ‘Combinatorics of the tropical Torelli map’, Algebra Number Theory 6(6) (2012), 1133–1169.CrossRefGoogle Scholar
Chan, M., ‘Topology of the tropical moduli spaces $M_{2,n}$’, Preprint, 2015, arXiv:1507.03878 [math].Google Scholar
Chan, M., ‘Lectures on tropical curves and their moduli spaces’, Preprint, 2016, arXiv:1606.02778 [math], Proceedings of the School on Moduli of Curves, Guanajuato, Lecture Notes of the Unione Matematica Italiana, Springer-UMI, to appear.CrossRefGoogle Scholar
Cavalieri, R., Hampe, S., Markwig, H. and Ranganathan, D., ‘Moduli spaces of rational weighted stable curves and tropical geometry’, Forum Math. Sigma 4 e9 (2016), 35.Google Scholar
Cavalieri, R., Markwig, H. and Ranganathan, D., ‘Tropicalizing the space of admissible covers’, Math. Ann. 364(3–4) 1275–1313.CrossRefGoogle Scholar
Chan, M., Melo, M. and Viviani, F., ‘Tropical Teichmüller and Siegel spaces’, inAlgebraic and Combinatorial Aspects of Tropical Geometry, Contemp. Math., vol. 589 (Amer. Math. Soc., Providence, RI, 2013), 45–85.CrossRefGoogle Scholar
Conrad, B. and Temkin, M., ‘Descent for non-archimedean analytic spaces’, Preprint, 2019, arXiv:1912.06230 [math].Google Scholar
Caporaso, L. and Viviani, F., ‘Torelli theorem for graphs and tropical curves’, Duke Math. J. 153(1) (2010), 129–171.CrossRefGoogle Scholar
Deligne, P. and Mumford, D., ‘The irreducibility of the space of curves of given genus’, Inst. Hautes Études Sci. Publ. Math. (36) (1969), 75–109.CrossRefGoogle Scholar
Francois, G. and Hampe, S., ‘Universal families of rational tropical curves’, Canad. J. Math. 65(1) (2013), 120–148.CrossRefGoogle Scholar
Foster, Tyler, Ranganathan, Dhruv, Talpo, Mattia and Ulirsch, Martin, ‘Logarithmic Picard groups, chip firing, and the combinatorial rank’, Math. Z. 291(1–2) (2019), 313–327.CrossRefGoogle Scholar
Giansiracusa, J. and Giansiracusa, N., ‘Equations of tropical varieties’, Duke Math. J. 165(18) (2016), 3379–3433.CrossRefGoogle Scholar
Gathmann, A., Kerber, M. and Markwig, H., ‘Tropical fans and the moduli spaces of tropical curves’, Compos. Math. 145(1) (2009), 173–195.CrossRefGoogle Scholar
Gathmann, A. and Markwig, H., ‘Kontsevich’s formula and the WDVV equations in tropical geometry’, Adv. Math. 217(2) (2008), 537–560.CrossRefGoogle Scholar
Gross, M., Tropical Geometry and Mirror Symmetry, CBMS Regional Conference Series in Mathematics, vol. 114, (Published for the Conference Board of the Mathematical Sciences, Washington, DC, 2011), by the American Mathematical Society, Providence, RI.CrossRefGoogle Scholar
Gross, A., ‘Intersection theory on tropicalizations of toroidal embeddings’, Proc. Lond. Math. Soc. (3) 116(6) (2018), 1365–1405.CrossRefGoogle Scholar
Geraschenko, A. and Satriano, M., ‘Toric stacks I: The theory of stacky fans’, Trans. Am. Math. Soc. 367(2) (2015), 1033–1071.CrossRefGoogle Scholar
Huszar, A., Marcus, S. and Ulirsch, M., ‘Clutching and gluing in tropical and logarithmic geometry’, J. Pure Appl. Algebra 223(5) (2019), 2036–2061.CrossRefGoogle Scholar
Kato, K., ‘Logarithmic structures of Fontaine-Illusie’, inAlgebraic Analysis, Geometry, and Number Theory (Baltimore, MD, 1988) (Johns Hopkins University Press, Baltimore, MD, 1989), 191–224.Google Scholar
Kato, K., ‘Toric singularities’, Amer. J. Math. 116(5) (1994), 1073–1099.CrossRefGoogle Scholar
Kato, F., ‘Exactness, integrality, and log modifications’, Preprint, July 1999, arXiv:9907124.Google Scholar
Kato, F., ‘Log smooth deformation and moduli of log smooth curves’, Internat. J. Math. 11(2) (2000), 215–232.CrossRefGoogle Scholar
Kempf, G., Knudsen, F. F., Mumford, D. and Saint-Donat, B., Toroidal Embeddings. I, Lecture Notes in Mathematics, Vol. 339, (Springer, Berlin, 1973).CrossRefGoogle Scholar
Kato, K. and Nakayama, C., ‘Log Betti cohomology, log étale cohomology, and log de Rham cohomology of log schemes over C’, Kodai Math. J. 22(2) (1999), 161–186.CrossRefGoogle Scholar
Knudsen, F. F., ‘The projectivity of the moduli space of stable curves. II. The stacks M g, n’, Math. Scand. 52(2) (1983), 161–199.CrossRefGoogle Scholar
Lorscheid, O., ‘Scheme theoretic tropicalization’, Preprint, 2015, arXiv:1508.07949 [math].Google Scholar
Metzler, D., ‘Topological and Smooth Stacks’, Preprint, 2003, arXiv:math/0306176.Google Scholar
Mikhalkin, G., Tropical geometry and its applications, International Congress of Mathematicians. Vol. II (Eur. Math. Soc., Zürich, 2006), 827–852.Google Scholar
Mikhalkin, G., ‘Moduli spaces of rational tropical curves’, inProceedings of Gökova Geometry–Topology Conference, 2006, Gökova Geometry/Topology Conference (GGT), Gökova (2007), 39–51.Google Scholar
Molcho, S., Universal Stacky Semistable Resolution. Preprint, 2014, arXiv:1601.00302 [math].Google Scholar
Maclagan, D. and Rincón, F., ‘Tropical schemes, tropical cycles, and valuated matroids’, Preprint, 2014, arXiv:1401.4654 [math].Google Scholar
Maclagan, D. and Rincón, F., ‘Tropical ideals’, Compos. Math. 154(3) (2018), 640–670.CrossRefGoogle Scholar
Molcho, S. and Wise, J., ‘The logarithmic Picard group and its tropicalization’, Preprint, 2018, arXiv:1807.11364 [math].Google Scholar
Noohi, B., ‘Foundations of Topology Stacks I’, Preprint, 2005, arXiv:math/0503247.Google Scholar
Nishinou, T. and Siebert, B., ‘Toric degenerations of toric varieties and tropical curves’, Duke Math. J. 135(1) (2006), 1–51.CrossRefGoogle Scholar
Olsson, M. C., ‘Logarithmic geometry and algebraic stacks’, Ann. Sci. Éc. Norm. Supér. (4) 36(5) (2003), 747–791.CrossRefGoogle Scholar
Porta, M. and Yu, T. Y., ‘Higher analytic stacks and GAGA theorems’, Adv. Math. 302 (2016), 351–409.CrossRefGoogle Scholar
Ranganathan, D., ‘Skeletons of stable maps I: rational curves in toric varieties’, J. Lond. Math. Soc. (2) 95(3) (2017), 804–832.CrossRefGoogle Scholar
Raynaud, M. and Gruson, L., ‘Critères de platitude et de projectivité. Techniques de “platification” d’un module’, Invent. Math. 13 (1971), 1–89.CrossRefGoogle Scholar
Rosales, J. C. and Sánchez, P. A. G. a, Finitely generated commutative monoids, (Nova Science Publishers Inc., Commack, NY, 1999).Google Scholar
Ranganathan, D., Santos-Parker, K. and Wise, J., ‘Moduli of stable maps in genus one and logarithmic geometry I’, Geom. Topology 23 (2019), 3315–3366.CrossRefGoogle Scholar
Ranganathan, D., Santos-Parker, K. and Wise, J., ‘Moduli of stable maps in genus one and logarithmic geometry II’, Algebra Number Theory 13(8) (2019), 1765–1805.CrossRefGoogle Scholar
Serre, J.-P., Arbres, amalgames, $SL_{2}$. Rédigé avec la collaboration de Hyman Bass. Astérisque, No. 46. Société Mathématique de France, Paris, 1977.Google Scholar
Simpson, C., ‘Algebraic (geometric) $n$-stacks’, Preprint, 1996, arXiv:alg-geom/9609014.Google Scholar
The Stacks Project Authors, stacks project, http://stacks.math.columbia.edu, 2016.Google Scholar
Thuillier, A., ‘Géométrie toroïdale et géométrie analytique non archimédienne. Application au type d’homotopie de certains schémas formels’, Manuscripta Math. 123(4) (2007), 381–451.CrossRefGoogle Scholar
Toën, B. and Vaquié, M., ‘Algébrisation des variétés analytiques complexes et catégories dérivées’, Math. Ann. 342(4) (2008), 789–831.Google Scholar
Toën, B. and Vezzosi, G., ‘Homotopical algebraic geometry. II. Geometric stacks and applications’, Mem. Amer. Math. Soc. 193(902)x+224 (2008).Google Scholar
Ulirsch, M., ‘Tropical geometry of moduli spaces of weighted stable curves’, J. Lond. Math. Soc. (2) 92(2) (2015), 427–450.CrossRefGoogle Scholar
Ulirsch, M., ‘Functorial tropicalization of logarithmic schemes: the case of constant coefficients’, Proc. Lond. Math. Soc. (3) 114(6) (2017), 1081–1113.CrossRefGoogle Scholar
Ulirsch, M., ‘Tropicalization is a non-Archimedean analytic stack quotient’, Math. Res. Lett. 24(4) (2017), 1205–1237.CrossRefGoogle Scholar
Ulirsch, M., ‘Non-Archimedean geometry of Artin fans’, Adv. Math. 345 (2019), 346–381.CrossRefGoogle Scholar
Vistoli, A., ‘Intersection theory on algebraic stacks and on their moduli spaces’, Invent. Math. 97(3) (1989), 613–670.CrossRefGoogle Scholar
Viviani, Filippo, ‘Tropicalizing vs. compactifying the Torelli morphism’, inTropical and non-Archimedean Geometry, Contemp. Math., vol. 605 (American Mathematical Society, Providence, RI, 2013), 181–210.CrossRefGoogle Scholar
Yu, T. Y., ‘Tropicalization of the moduli space of stable maps’, Math. Z. 281(3–4) 1035–1059.CrossRefGoogle Scholar
Yu, T. Y., ‘Gromov compactness in non-archimedean analytic geometry’, J. Reine Angew. Math. 741 (2018), 179–210.CrossRefGoogle Scholar