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The moduli space of the tropicalizations of Riemann surfaces

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In this paper we study the moduli space of the tropicalizations of Riemann surfaces. We first tropicalize a smooth pointed Riemann surface by its pair of pants decomposition. Then we can construct the moduli space of tropicalizations based on a fixed regular tropicalization, and compactify it by adding strata parametrizing weighted contractions. This, in the covering level, is analogous to adding the frontier set, subordinate to a pants decomposition, to the Teichmüller space. We show that this compact moduli space is also Hausdorff. In the end, we compare this moduli space with the moduli space of Riemann surfaces, establishing a partial order-preserving correspondence between the stratifications of these two moduli spaces.

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References

  1. Abikoff, W.: Degenerating families of Riemann surfaces. Ann. Math. 2(105), 29–44 (1977)

    Article  MathSciNet  MATH  Google Scholar 

  2. Abramovich, D.: Moduli of algebraic and tropical curves. In: Colloquium De Giorgi 2010–2012, volume 4 of Colloquia, pp. 35–47. Ed. Norm., Pisa (2013)

  3. Abramovich, D., Caporaso, L., Payne, S.: The tropicalization of the moduli space of curves. Ann. Sci. Éc. Norm. Supér. (4) 48(4), 765–809 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  4. Arbarello, E., Cornalba, M., Griffiths, P.A.: Geometry of algebraic curves. Volume II. With a contribution by Joseph Daniel Harris, volume 268 of Grundlehren der Mathematischen Wissenschaften. Springer, Berlin (2011)

  5. Bers, L.: Spaces of degenerating Riemann surfaces. In: Discontinuous Groups and Riemann Surfaces (Proc. Conf., Univ. Maryland, College Park, Md., 1973), pp. 43–55. Annals of Mathematics Studies, No. 79 (1974)

  6. Brannetti, S., Melo, M., Viviani, F.: On the tropical Torelli map. Adv. Math. 226(3), 2546–2586 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  7. Caporaso, L.: Algebraic and tropical curves: comparing their moduli spaces. In: Handbook of Moduli, vol. I, pp. 119–160. International Press; Beijing: Higher Education Press, Somerville, MA (2015)

  8. Caporaso, L.: Tropical methods in the moduli theory of algebraic curves. In: Algebraic Geometry: Salt Lake City 2015. 2015 Summer Research Institute in Algebraic Geometry, University of Utah, Salt Lake City, UT, USA, July 13–31, 2015. Proceedings. Part 2, pp. 103–138. American Mathematical Society (AMS), Providence, RI; Clay Mathematics Institute, Cambridge, MA (2018)

  9. Chan, M., Galatius, S., Payne, S.: Tropical curves, graph complexes, and top weight cohomology of \({\cal{M}} _g\). J. Am. Math. Soc. 34(2), 565–594 (2021)

    Article  MATH  Google Scholar 

  10. Chan, M., Galatius, S., Payne, S.: Topology of moduli spaces of tropical curves with marked points. In: Facets of Algebraic Geometry: A Collection in Honour of William Fulton’s 80th Birthday. Cambridge University Press (2022). Preprint arXiv:1903.07187

  11. Looijenga, E.J.N.: Moduli spaces of Riemann surfaces. In: Introduction to Moduli Spaces of Riemann Surfaces and Tropical Curves, volume 14 of Surv. Mod. Math., pp. 1–87. International Press, Somerville, MA; Higher Education Press, Beijing (2017)

  12. Maclagan, D., Sturmfels, B.: Introduction to Tropical Geometry. Graduate Studies in Mathematics, vol. 161. American Mathematical Society (AMS), Providence (2015)

    MATH  Google Scholar 

  13. Odaka, Y.: Tropical geometric compactification of moduli, \({\rm I}- M_g\) case. In: Moduli of \(K\)-Stable Varieties, pp. 75–101. Springer, Cham (2019)

  14. Wolpert, S.A.: Geometry of the Weil–Petersson completion of Teichmüller space. In: Surveys in Differential Geometry. Volume VIII: Lectures on Geometry and Topology Held in Honor of Calabi, Lawson, Siu, and Uhlenbeck at Harvard University, Cambridge, MA, USA, May 3–5, 2002, pp. 357–393. International Press, Somerville, MA (2003)

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Acknowledgements

I would like to thank the referee for the helpful comments and suggestions which certainly improved the quality of the paper.

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  1. School of Mathematics, Tata Institute of Fundamental Research, Homi Bhabha Road, Mumbai, 400005, India

    Dali Shen

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  1. Dali Shen
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Correspondence to Dali Shen.

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Shen, D. The moduli space of the tropicalizations of Riemann surfaces. manuscripta math. 171, 397–422 (2023). https://doi.org/10.1007/s00229-022-01383-1

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