Advertisement

Towards a Tropical Hodge Bundle

  • Bo Lin
  • Martin UlirschEmail author
Chapter
Part of the Fields Institute Communications book series (FIC, volume 80)

Abstract

The moduli space \(\mathop{\mathrm{M}}\nolimits _{g}^{\text{trop}}\) of tropical curves of genus g is a generalized cone complex that parametrizes metric vertex-weighted graphs of genus g. For each such graph Γ, the associated canonical linear system | K Γ | has the structure of a polyhedral complex. In this article, we propose a tropical analogue of the Hodge bundle on \(\mathop{\mathrm{M}}\nolimits _{g}^{\text{trop}}\) and study its basic combinatorial properties. Our construction is illustrated with explicit computations and examples.

MSC 2010 codes:

14T05 

Notes

Acknowledgements

This article was initiated during the Apprenticeship Weeks (22 August–2 September 2016), led by Bernd Sturmfels, as part of the Combinatorial Algebraic Geometry Semester at the Fields Institute. Both authors would like to acknowledge his input. Thanks are also due to the Max-Planck-Institute of Mathematics in the Sciences in Leipzig, Germany, for its hospitality. The second author would like to thank Diane Maclagan for several discussions related to the topic of this note, as well as the Fields Institute for Research in Mathematical Sciences. Finally, many thanks are due to the anonymous referees for several helpful comments and suggestions.

References

  1. 1.
    Dan Abramovich, Lucia Caporaso, and Sam Payne: The tropicalization of the moduli space of curves, Ann. Sci. Éc. Norm. Supér. (4) 48 (2015) 765–809.Google Scholar
  2. 2.
    Omid Amini and Matthew Baker: Linear series on metrized complexes of algebraic curves, Math. Ann. 362 (2015) 55–106.Google Scholar
  3. 3.
    Omid Amini and Lucia Caporaso: Riemann-Roch theory for weighted graphs and tropical curves, Adv. Math. 240 (2013) 1–23.Google Scholar
  4. 4.
    Matt Bainbridge, Dawei Chen, Quentin Gendron, Samuel Grushevsky, and Martin Möller: Compactifications of strata of abelian differentials, arXiv:1604.08834 [math.AG].Google Scholar
  5. 5.
    Matthew Baker: Specialization of linear systems from curves to graphs, Algebra Number Theory 2 (2008) 613–653.Google Scholar
  6. 6.
    Matthew Baker and Serguei Norine: Riemann-Roch and Abel-Jacobi on a finite graph, Adv. Math. 215 (2007) 766–788.Google Scholar
  7. 7.
    Silvia Brannetti, Margarida Melo, and Filippo Viviani, Filippo: On the tropical Torelli map, Adv. Math. 226 (2011) 2546–2586.Google Scholar
  8. 8.
    Lucia Caporaso: Algebraic and tropical curves: comparing their moduli spaces, in Handbook of moduli I, 119–160, Adv. Lect. Math. 24, Int. Press, Somerville, MA, 2013.Google Scholar
  9. 9.
    _________ : Tropical methods in the moduli theory of algebraic curves, arXiv:1606.00323 [math.AG].Google Scholar
  10. 10.
    Lucia Caporaso and Filippo Viviani: Torelli theorem for graphs and tropical curves, Duke Math. J. 153 (2010) 129–171.Google Scholar
  11. 11.
    Renzo Cavalieri, Simon Hampe, Hannah Markwig, and Dhruv Ranganathan: Moduli spaces of rational weighted stable curves and tropical geometry, Forum Math. Sigma 4 (2016) e9 35pp.Google Scholar
  12. 12.
    Renzo Cavalieri, Hannah Markwig, and Dhruv Ranganathan: Tropicalizing the space of admissible covers, Math. Ann. 364 (2016) 1275–1313.Google Scholar
  13. 13.
    Melody Chan: Combinatorics of the tropical Torelli map, Algebra Number Theory 6 (2012) 1133–1169.Google Scholar
  14. 14.
    _________ : Topology of the tropical moduli spaces M 2,n, arXiv:1507.03878 [math.CO].Google Scholar
  15. 15.
    Melody Chan, Soren Galatius, and Sam Payne: The tropicalization of the moduli space of curves II: Topology and applications, arXiv:1604.03176 [math.AG].Google Scholar
  16. 16.
    Melody Chan, Margarida Melo, and Filippo Viviani: Tropical Teichmüller and Siegel spaces, in Algebraic and combinatorial aspects of tropical geometry, 45–85, Contemp. Math. 589, American Mathematical Society, Providence, RI, 2013.Google Scholar
  17. 17.
    David Eisenbud and Joe Harris: Limit linear series: basic theory, Invent. Math. 85 (1986) 337–371.Google Scholar
  18. 18.
    Andreas Gathmann and Michael Kerber: A Riemann-Roch theorem in tropical geometry, Math. Z. 259 (2008) 217–230.Google Scholar
  19. 19.
    Andreas Gathmann, and Michael Kerber, and Hannah Markwig: Tropical fans and the moduli spaces of tropical curves, Compos. Math. 145 (2009) 173–195.Google Scholar
  20. 20.
    Andreas Gathmann and Hannah Markwig: Kontsevich’s formula and the WDVV equations in tropical geometry, Adv. Math. 217 (2008) 537–560.Google Scholar
  21. 21.
    Mark Gross: Tropical geometry and mirror symmetry, CBMS Regional Conference Series in Mathematics 114, American Mathematical Society, Providence, RI, 2011.Google Scholar
  22. 22.
    Christian Haase, Gregg Musiker,and Josephine Yu: Linear systems on tropical curves, Math. Z. 270 (2012) 1111–1140.Google Scholar
  23. 23.
    Bo Lin: Computing linear systems on metric graphs, Journal of Symbolic Compututation (to appear).Google Scholar
  24. 24.
    Grigory Mikhalkin: Tropical geometry and its applications, in International Congress of Mathematicians II, 827–852, Eur. Math. Soc., Zürich, 2006.Google Scholar
  25. 25.
    _________ : Moduli spaces of rational tropical curves, in Proceedings of Gökova Geometry-Topology Conference 2006, 39–51, Gökova Geometry/Topology Conference, Gökova, 2007.Google Scholar
  26. 26.
    Grigory Mikhalkin and Ilia Zharkov: Tropical curves, their Jacobians and theta functions, in Curves and abelian varieties, 203–230, Contemp. Math. 465, American Mathematical Society, Providence, RI, 2008.Google Scholar
  27. 27.
    Brian Osserman: Limit linear series for curves not of compact type, arXiv:1406.6699 [math.AG].Google Scholar
  28. 28.
    Dhruv Ranganathan: Skeletons of stable maps I: Rational curves in toric varieties, J. Lond. Math. Soc. (2) 95(3) (2017) 804–832.Google Scholar
  29. 29.
    Martin Ulirsch: Functorial tropicalization of logarithmic schemes: The case of constant coefficients, Proc. Lond. Math. Soc. (3) 114(6) (2017) 1081–1113.Google Scholar
  30. 30.
    _________ : Tropical geometry of moduli spaces of weighted stable curves, J. Lond. Math. Soc. (2) 92 (2015) 427–450.Google Scholar
  31. 31.
    Ravi Vakil: The moduli space of curves and its tautological ring, Notices Amer. Math. Soc. 50 (2003) 647–658.Google Scholar
  32. 32.
    Filippo Viviani: Tropicalizing vs. compactifying the Torelli morphism, in Tropical and non-Archimedean geometry, 181–210, Contemp. Math. 605, American Mathematical Society, Providence, RI, 2013.Google Scholar
  33. 33.
    Tony Yue Yu: Tropicalization of the moduli space of stable maps, Math. Z. 281 (2015) 1035–1059.Google Scholar

Copyright information

© Springer Science+Business Media LLC 2017

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Texas at AustinAustinUSA
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA

Personalised recommendations