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Topology of tropical moduli spaces of weighted stable curves in higher genus

  • Siddarth Kannan , Shiyue Li EMAIL logo , Stefano Serpente and Claudia He Yun
From the journal Advances in Geometry


We study the topology of moduli spaces of weighted stable tropical curves Δg,w with fixed genus and unit volume. The space Δg,w arises as the dual complex of the divisor of singular curves in Hassett’s moduli space Mg,w of weighted stable curves. When the genus is positive, we show that Δg,w is simply connected for any choice of weight vector w. We also give a formula for the Euler characteristic of Δg,w in terms of the combinatorics of the weight vector.

MSC 2010: 14T20

Funding statement: SK was supported by an NSF Graduate Research Fellowship.


We are grateful to Melody Chan for suggesting a motivic approach to the proof of Theorem 1.2, to Sam Payne for help with questions about mixed Hodge structures and connections to geometric group theory, and to Sam Freedman for many useful conversations related to this work. We also thank the organizers of the 2020 Summer Tropical Algebraic Geometry Online SeminAUR (STAGOSAUR)/Algebraic and Tropical Online Meetings (ATOM) for great learning opportunities and for bringing us together.

  1. Communicated by: R. Cavalieri


[1] D. Allcock, D. Corey, S. Payne, Tropical moduli spaces as symmetric Delta-complexes. Bull. Lond. Math. Soc. 54 (2022), 193–205. MR439693110.1112/blms.12570Search in Google Scholar

[2] T. M. Apostol, Introduction to analytic number theory. Springer 1976. MR0434929 Zbl 0335.1000110.1007/978-1-4757-5579-4Search in Google Scholar

[3] F. Ardila, C. J. Klivans, The Bergman complex of a matroid and phylogenetic trees. J. Combin. Theory Ser. B 96 (2006), 38–49. MR2185977 Zbl 1082.0502110.1016/j.jctb.2005.06.004Search in Google Scholar

[4] K. Behrend, Cohomology of stacks. In: Intersection theory and moduli, 249–294, Abdus Salam Int. Cent. Theoret. Phys., Trieste 2004. MR2172499 Zbl 1081.58003Search in Google Scholar

[5] G. Bini, J. Harer, Euler characteristics of moduli spaces of curves. J. Eur. Math. Soc. 13 (2011), 487–512. MR2746773 Zbl 1227.1402810.4171/JEMS/259Search in Google Scholar

[6] R. Cavalieri, S. Hampe, H. Markwig, D. Ranganathan, Moduli spaces of rational weighted stable curves and tropical geometry. Forum Math. Sigma 4 (2016), Paper No. e9, 35 pages. MR3507917 Zbl 1373.1406310.1017/fms.2016.7Search in Google Scholar

[7] A. Cerbu, S. Marcus, L. Peilen, D. Ranganathan, A. Salmon, Topology of tropical moduli of weighted stable curves. Adv. Geom. 20 (2020), 445–462. MR4160280 Zbl 1460.1414510.1515/advgeom-2019-0034Search in Google Scholar

[8] M. Chan, Topology of the tropical moduli spaces Δ2,nBeitr. Algebra Geom. 63 (2022), 69–93. MR4393951 Zbl 0749560210.1007/s13366-021-00563-6Search in Google Scholar

[9] M. Chan, C. Faber, S. Galatius, S. Payne, The Sn-equivariant top weight Euler characteristic of Mgn To appear in Amer. J. Math.Search in Google Scholar

[10] M. Chan, S. Galatius, S. Payne, Tropical curves, graph complexes, and top weight cohomology ofMgJ. Amer. Math. Soc. 34 (2021), 565–594. MR4280867 Zbl 1485.1404510.1090/jams/965Search in Google Scholar

[11] M. Chan, S. Galatius, S. Payne, Topology of moduli spaces of tropical curves with marked points. In: Facets of algebraic geometry. Vol. I, volume 472 of London Math. Soc. Lecture Note Ser., 77–131, Cambridge Univ. Press 2022. MR4381898 Zbl 1489.1403810.1017/9781108877831.004Search in Google Scholar

[12] A. Craw, An introduction to motivic integration. In: Strings and geometry, volume 3 of Clay Math. Proc., 203–225, Amer. Math. Soc. 2004. MR2103724 Zbl 1156.14307Search in Google Scholar

[13] M. Culler, K. Vogtmann, Moduli of graphs and automorphisms of free groups. Invent. Math. 84 (1986), 91–119. MR830040 Zbl 0589.2002210.1007/BF01388734Search in Google Scholar

[14] P. Deligne, Théorie de Hodge. III. Inst. Hautes Études Sci. Publ. Math. no. 44 (1974), 5–77. MR498552 Zbl 0237.1400310.1007/BF02685881Search in Google Scholar

[15] D. Edidin, Equivariant geometry and the cohomology of the moduli space of curves. In: Handbook of moduli. Vol. I, volume 24 of Adv. Lect. Math., 259–292, Int. Press, Somerville, MA 2013. MR3184166 Zbl 1322.14003Search in Google Scholar

[16] J. L. Harer, The virtual cohomological dimension of the mapping class group of an orientable surface. Invent. Math. 84 (1986), 157–176. MR830043 Zbl 0592.5700910.1007/BF01388737Search in Google Scholar

[17] W. J. Harvey, Boundary structure of the modular group. In: Riemann surfaces and related topics: Proceedings of the 1978 Stony Brook Conference (State Univ. New York, Stony Brook, N.Y., 1978), volume 97 of Ann. of Math. Stud., 245–251, Princeton Univ. Press 1981. MR624817 Zbl 0461.3003610.1515/9781400881550-019Search in Google Scholar

[18] B. Hassett, Moduli spaces of weighted pointed stable curves. Adv. Math. 173 (2003), 316–352. MR1957831 Zbl 1072.1401410.1016/S0001-8708(02)00058-0Search in Google Scholar

[19] A. Hatcher, Homological stability for automorphism groups of free groups. Comment. Math. Helv. 70 (1995), 39–62. MR1314940 Zbl 0836.5700310.1007/BF02565999Search in Google Scholar

[20] S. Kannan, S. Li, S. Serpente, C.-h. Yun, Topology of tropical moduli spaces of weighted stable curves in higher genus. Preprint 2020, arXiv:2010.11767 [math.CO]Search in Google Scholar

[21] T. Komatsu, K. Liptai, I. Mező, Incomplete poly-Bernoulli numbers associated with incomplete Stirling numbers. Publ. Math. Debrecen 88 (2016), 357–368. MR3491746 Zbl 1389.1105410.5486/PMD.2016.7361Search in Google Scholar

[22] F. Loeser, Seattle lectures on motivic integration. In: Algebraic geometry—Seattle 2005 Part 2, volume 80 of Proc. Sympos. Pure Math., 745–784, Amer. Math. Soc. 2009. MR2483954 Zbl 1181.1401710.1090/pspum/080.2/2483954Search in Google Scholar

[23] C. A. M. Peters, J. H. M. Steenbrink, Mixed Hodge structures. Springer 2008. MR2393625 Zbl 1138.14002Search in Google Scholar

[24] A. Robinson, S. Whitehouse, The tree representation of Σn+1J. Pure Appl. Algebra 111 (1996), 245–253. MR1394355 Zbl 0865.5501010.1016/0022-4049(95)00116-6Search in Google Scholar

[25] M. Ulirsch, Tropical geometry of moduli spaces of weighted stable curves. J. Lond. Math. Soc. (2) 92 (2015), 427–450. MR3404032 Zbl 1349.1419710.1112/jlms/jdv032Search in Google Scholar

[26] K. Vogtmann, Local structure of some Out(Fn-complexes. Proc. Edinburgh Math. Soc. (2) 33 (1990), 367–379. MR1077791 Zbl 0694.2002110.1017/S0013091500004818Search in Google Scholar

Published Online: 2023-08-11
Published in Print: 2023-08-28

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