Tropical ψ classes

Cavalieri, Renzo¹ ; Gross, Andreas² ; Markwig, Hannah³

¹ Department of Mathematics, Colorado State University, Fort Collins, CO, United States
² Department of Mathematics, Colorado State University, Fort Collins, CO, United States, Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany
³ Fachbereich Mathematik, Eberhard Karls Universität Tübingen, Tübingen, Germany

Geometry & topology, Tome 26 (2022) no. 8, pp. 3421-3524.

Résumé

We introduce a tropical geometric framework that allows us to define $\psi$ classes for moduli spaces of tropical curves of arbitrary genus. We prove correspondence theorems between algebraic and tropical $\psi$ classes for some one-dimensional families of genus-one tropical curves.

Détail

Keywords: tropical geometry, moduli space of curves, tautological classes, psi classes, intersection theory

Bibliographie

[1] D Abramovich, L Caporaso, S Payne, The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér. 48 (2015) 765 | DOI

[2] L Allermann, J Rau, First steps in tropical intersection theory, Math. Z. 264 (2010) 633 | DOI

[3] E Arbarello, M Cornalba, Combinatorial and algebro-geometric cohomology classes on the moduli spaces of curves, J. Algebraic Geom. 5 (1996) 705

[4] P Belmans, A J De Jong, Others, The Stacks project, electronic reference (2005–)

[5] B Bertrand, E Brugallé, G Mikhalkin, Tropical open Hurwitz numbers, Rend. Semin. Mat. Univ. Padova 125 (2011) 157 | DOI

[6] L J Billera, S P Holmes, K Vogtmann, Geometry of the space of phylogenetic trees, Adv. in Appl. Math. 27 (2001) 733 | DOI

[7] E Brugallé, I Itenberg, G Mikhalkin, K Shaw, Brief introduction to tropical geometry, from: "Proceedings of the Gökova Geometry–Topology Conference 2014" (editors S Akbulut, D Auroux, T Önder), GGT (2015) 1

[8] E Brugallé, G Mikhalkin, Floor decompositions of tropical curves: the planar case, from: "Proceedings of Gökova Geometry–Topology Conference 2008" (editors S Akbulut, T Önder, R J Stern), GGT (2009) 64

[9] L Caporaso, Algebraic and tropical curves: comparing their moduli spaces, from: "Handbook of moduli, I" (editors G Farkas, I Morrison), Adv. Lect. Math. 24, International (2013) 119

[10] L Caporaso, Gonality of algebraic curves and graphs, from: "Algebraic and complex geometry" (editors A Frühbis-Krüger, R N Kloosterman, M Schütt), Springer Proc. Math. Stat. 71, Springer (2014) 77 | DOI

[11] L Caporaso, Recursive combinatorial aspects of compactified moduli spaces, from: "Proceedings of the International Congress of Mathematicians" (editors B Sirakov, P N d Souza, M Viana), World Sci. (2018) 635

[12] D Cartwright, Combinatorial tropical surfaces, preprint (2015)

[13] R Cavalieri, M Chan, M Ulirsch, J Wise, A moduli stack of tropical curves, Forum Math. Sigma 8 (2020) | DOI

[14] R Cavalieri, H Markwig, D Ranganathan, Tropicalizing the space of admissible covers, Math. Ann. 364 (2016) 1275 | DOI

[15] S Flossmann, Tropical intersection theory on Rn, preprint (2014)

[16] S Fomin, G Mikhalkin, Labeled floor diagrams for plane curves, J. Eur. Math. Soc. 12 (2010) 1453 | DOI

[17] G François, J Rau, The diagonal of tropical matroid varieties and cycle intersections, Collect. Math. 64 (2013) 185 | DOI

[18] G Francois, S Hampe, Universal families of rational tropical curves, Canad. J. Math. 65 (2013) 120 | DOI

[19] W Fulton, B Sturmfels, Intersection theory on toric varieties, Topology 36 (1997) 335 | DOI

[20] A Gathmann, M Kerber, H Markwig, Tropical fans and the moduli spaces of tropical curves, Compos. Math. 145 (2009) 173 | DOI

[21] A Gathmann, H Markwig, Kontsevich’s formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008) 537 | DOI

[22] E Gawrilow, M Joswig, polymake: a framework for analyzing convex polytopes, from: "Polytopes: combinatorics and computation" (editors G Kalai, G M Ziegler), DMV Sem. 29, Birkhäuser (2000) 43 | DOI

[23] A Gibney, D Maclagan, Equations for Chow and Hilbert quotients, Algebra Number Theory 4 (2010) 855 | DOI

[24] C Goldner, Counting tropical rational curves with cross-ratio constraints, Math. Z. 297 (2021) 133 | DOI

[25] A Gross, Correspondence theorems via tropicalizations of moduli spaces, Commun. Contemp. Math. 18 (2016) | DOI

[26] A Gross, Intersection theory on tropicalizations of toroidal embeddings, Proc. Lond. Math. Soc. 116 (2018) 1365 | DOI

[27] A Gross, F Shokrieh, A sheaf-theoretic approach to tropical homology, preprint (2019)

[28] M Gross, B Siebert, Mirror symmetry via logarithmic degeneration data, I, J. Differential Geom. 72 (2006) 169

[29] S Hampe, a-tint : a polymake extension for algorithmic tropical intersection theory, European J. Combin. 36 (2014) 579 | DOI

[30] I Itenberg, L Katzarkov, G Mikhalkin, I Zharkov, Tropical homology, Math. Ann. 374 (2019) 963 | DOI

[31] P Jell, J Rau, K Shaw, Lefschetz (1,1)–theorem in tropical geometry, Épijournal Géom. Algébrique 2 (2018) | DOI

[32] Z. Jin, Tropical intersection theory on moduli stack of curve coverings, PhD thesis, University of Cambridge (2019)

[33] M M Kapranov, Chow quotients of Grassmannians, I, from: "I M Gelfand Seminar" (editors S Gelfand, S Gindikin), Adv. Soviet Math. 16, Amer. Math. Soc. (1993) 29

[34] E Katz, Tropical intersection theory from toric varieties, Collect. Math. 63 (2012) 29 | DOI

[35] M Kerber, H Markwig, Counting tropical elliptic plane curves with fixed j–invariant, Comment. Math. Helv. 84 (2009) 387 | DOI

[36] M Kerber, H Markwig, Intersecting Psi-classes on tropical M0,n, Int. Math. Res. Not. 2009 (2009) 221 | DOI

[37] M Kontsevich, Enumeration of rational curves via torus actions, from: "The moduli space of curves" (editors R Dijkgraaf, C Faber, G van der Geer), Progr. Math. 129, Birkhäuser (1995) 335 | DOI

[38] G Mikhalkin, Enumerative tropical algebraic geometry in R2, J. Amer. Math. Soc. 18 (2005) 313 | DOI

[39] G Mikhalkin, Tropical geometry and its applications, from: "International Congress of Mathematicians" (editors B Sirakov, P N d Souza, M Viana), Eur. Math. Soc. (2006) 827

[40] G Mikhalkin, Moduli spaces of rational tropical curves, from: "Proceedings of Gökova Geometry–Topology Conference 2006" (editors S Akbulut, T Önder, R J Stern), GGT (2007) 39

[41] G. Mikhalkin, J. Rau, Tropical geometry, book project (2020)

[42] G Mikhalkin, I Zharkov, Tropical curves, their Jacobians and theta functions, from: "Curves and abelian varieties" (editors V Alexeev, A Beauville, C H Clemens, E Izadi), Contemp. Math. 465, Amer. Math. Soc. (2008) 203 | DOI

[43] G Mikhalkin, I Zharkov, Tropical eigenwave and intermediate Jacobians, from: "Homological mirror symmetry and tropical geometry" (editors R Castaño-Bernard, F Catanese, M Kontsevich, T Pantev, Y Soibelman, I Zharkov), Lect. Notes Unione Mat. Ital. 15, Springer (2014) 309 | DOI

[44] D Mumford, Towards an enumerative geometry of the moduli space of curves, from: "Arithmetic and geometry, II" (editors M Artin, J Tate), Progr. Math. 36, Birkhäuser (1983) 271 | DOI

[45] D Ranganathan, Skeletons of stable maps, II : superabundant geometries, Res. Math. Sci. 4 (2017) | DOI

[46] D Ranganathan, K Santos-Parker, J Wise, Moduli of stable maps in genus one and logarithmic geometry, I, Geom. Topol. 23 (2019) 3315 | DOI

[47] D Ranganathan, K Santos-Parker, J Wise, Moduli of stable maps in genus one and logarithmic geometry, II, Algebra Number Theory 13 (2019) 1765 | DOI

[48] J Rau, Intersections on tropical moduli spaces, Rocky Mountain J. Math. 46 (2016) 581 | DOI

[49] K M Shaw, A tropical intersection product in matroidal fans, SIAM J. Discrete Math. 27 (2013) 459 | DOI

[50] K Shaw, Tropical surfaces, preprint (2015)

[51] D E Speyer, Tropical geometry, PhD thesis, University of California, Berkeley (2005)

[52] D E Speyer, Parameterizing tropical curves, I : Curves of genus zero and one, Algebra Number Theory 8 (2014) 963 | DOI

[53] D Speyer, B Sturmfels, The tropical Grassmannian, Adv. Geom. 4 (2004) 389 | DOI

[54] J Tevelev, Compactifications of subvarieties of tori, Amer. J. Math. 129 (2007) 1087 | DOI

[55] C. Torchiani, Enumerative geometry of rational and elliptic tropical curves in Rm, PhD thesis, Universität Kaiserslautern (2014)

[56] I Tyomkin, Tropical geometry and correspondence theorems via toric stacks, Math. Ann. 353 (2012) 945 | DOI

[57] I Tyomkin, Enumeration of rational curves with cross-ratio constraints, Adv. Math. 305 (2017) 1356 | DOI

[58] M Ulirsch, Functorial tropicalization of logarithmic schemes: the case of constant coefficients, Proc. Lond. Math. Soc. 114 (2017) 1081 | DOI

[59] M Ulirsch, A non-Archimedean analogue of Teichmüller space and its tropicalization, Selecta Math. (N.S.) 27 (2021) | DOI

[60] R Vakil, The moduli space of curves and Gromov–Witten theory, from: "Enumerative invariants in algebraic geometry and string theory" (editors K Behrend, M Manetti), Lecture Notes in Math. 1947, Springer (2008) 143 | DOI

[61] E Witten, Two-dimensional gravity and intersection theory on moduli space, from: "Surveys in differential geometry" (editors H B Lawson Jr., S T Yau), Lehigh Univ. (1991) 243

[62] B Zimmermann, Finite groups of outer automorphisms of free groups, Glasgow Math. J. 38 (1996) 275 | DOI