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Combinatorics of tropical Hurwitz cycles

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Abstract

We study properties of the tropical double Hurwitz loci defined by Bertram, Cavalieri and Markwig. We show that all such loci are connected in codimension one. If we mark preimages of simple ramification points, then for a generic choice of such points, the resulting cycles are weakly irreducible, i.e. an integer multiple of an irreducible cycle. We study how Hurwitz cycles can be written as divisors of rational functions and show that they are numerically equivalent to a tropical version of a representation as a sum of boundary divisors. The results and counterexamples in this paper were obtained with the help of a-tint, an extension for polymake for tropical intersection theory.

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Notes

  1. In fact, one can consider this problem in even greater generality by counting covers \(C \rightarrow C'\), where C and \(C'\) are curves of prescribed genera g and \(g'\).

  2. See also https://github.com/simonhampe/atint.

  3. See also www.polymake.org.

References

  1. Allermann, L., Hampe, S., Rau, J.: On rational equivalence in tropical geometry (2014). arXiv:1408.1537 (preprint)

  2. Ardila, F., Klivans, C.: The Bergman complex of a matroid and phylogenetic trees. J. Comb. Theory Ser. B. 96, 38–49 (2006). arXiv:math/0311370v2

  3. Bertrand, B., Brugallée, E., Mikhalkin, G.: Tropical open Hurwitz numbers. Rend. Semin. Mat. Univ. Padova. 125, 157–171 (2011). arXiv:1005.4628

  4. Bertram, A., Cavalieri, R., Markwig, H.: Polynomiality, wall crossings and tropical geometry of rational double hurwitz cycles. JCTA 120, 1604–1631 (2013)

    Article  MathSciNet  Google Scholar 

  5. Berstein, I., Edmonds, A.L.: On the classification of generic branched coverings of surfaces. Ill. J. Math. 28(1), 64–82 (1984)

    MATH  MathSciNet  Google Scholar 

  6. Bogart, T., Jensen, A.N., Speyer, D., Sturmfels, B., Thomas, R.R.: Computing tropical varieties. J. Symb. Comput. 42(1–2), 54–73 (2007)

    Article  MATH  MathSciNet  Google Scholar 

  7. Clebsch, A.: Zur Theorie der Riemann’schen Fläche. Math. Ann. 6(2), 216–230 (1873)

    Article  MathSciNet  Google Scholar 

  8. Cavalieri, R., Johnson, P., Markwig, H.: Tropical Hurwitz numbers, J. Algebr. Comb. 32(2), 241–265 (2010). arXiv:0804.0579

  9. Cavalieri, R., Markwig, H., Ranganathan, D.: Tropical compactification and the gromov-witten theory of \(\mathbb{P}^1\) (2014). arXiv:1410.2837 (preprint)

  10. Cartwright, D., Payne, S.: Connectivity of tropicalizations. Math. Res. Lett. 19(5), 1089–1095 (2012). arXiv:1204.6589

  11. Ekedahl, T., Lando, S., Shapiro, M., Vainshtein, A.: Hurwitz numbers and intersections on moduli spaces of curves. Invent. Math. 146(2), 297–327 (2001)

    Article  MATH  MathSciNet  Google Scholar 

  12. Francois, G.: Cocycles on tropical varieties via piecewise polynomials. Proc. Am. Math. Soc. 141, 481–497 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  13. Francois, G., Rau, J.: The diagonal of tropical matroid varieties and cycle intersections. Collect. Math. 64, 185–210 (2013)

    Article  MATH  MathSciNet  Google Scholar 

  14. Feichtner, E.M., Sturmfels, B.: Matroid polytopes, nested sets and Bergman fans. Port. Math. Nova Série 62, 437–468 (2005)

    MATH  MathSciNet  Google Scholar 

  15. Fulton, W., Sturmfels, B.: Intersection theory on toric varieties. Topology 36(2), 335–353 (1997). (English)

    Article  MATH  MathSciNet  Google Scholar 

  16. Graber, T., Harris, J., Starr, J.: A note on Hurwitz schemes of covers of a positive genus curve (2002). arXiv:math/0205056 (preprint)

  17. Gawrilow, E., Joswig, M.: Polymake: a framework for analyzing convex polytopes. In: Polytopes: Combinatorics and Computation (Oberwolfach, 1997), pp. 43-73. (2000)

  18. Goulden, I., Jackson, D., Vakil, R.: Towards the geometry of double Hurwitz numbers. Adv. Math. 198(1), 43–92 (2005). arXiv:math/0309440

  19. Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli spaces of tropical curves. Compos. Math. 145, 173–195 (2009). arXiv:0708.2268

  20. Gibney, A., Maclagan, D.: Equations for Chow and Hilbert quotients. Algebra Number Theory. 7, 855–885 (2010). arXiv:0707.1801

  21. Graber, T., Vakil, R.: Relative virtual localization and vanishing of tautological classes on moduli spaces of curves. Duke Math. J. 130(1), 1–37 (2005). arXiv:math/0309227

  22. Hampe, S.: A-tint: a polymake extension for algorithmic tropical intersection theory. Eur. J. Comb. 36, 579–607 (2014). arXiv:1208.4248

  23. Hurwitz, A.: Über Riemann’sche Flächen mit gegebenen Verzweigungspunkten. Math. Ann. 39, 1–61 (1891)

    Article  MATH  MathSciNet  Google Scholar 

  24. Johnson, P.: Hurwitz numbers, ribbon graphs and tropicalizations. Contemp. Math. 580, 55–72 (2012). arXiv:1303.1543

  25. Kanev, V.: Irreducible components of Hurwitz spaces parameterizing Galois coverings of curves of positive genus. Pure Appl. Math. Q. 10(2), 193–222 (2014)

    Article  MATH  MathSciNet  Google Scholar 

  26. Kontsevich, M.: Intersection theory on the moduli space of curves and the matrix Airy function. Commun. Math. Phys. 147(1), 1–23 (1992)

    Article  MATH  MathSciNet  Google Scholar 

  27. Kerber, M., Markwig, H.: Intersecting psi-classes on tropical \(\cal M_{0,n}\). Int. Math. Res. Not. (2008). arXiv:0709.3953

  28. Mikhalkin, G.: Enumerative tropical algebraic geometry in \(\mathbb{R}^2\). J. Am. Math. Soc. 18(2), 313–377 (2005). arXiv:math/0312530

  29. Mikhalkin, G.: Moduli spaces of rational tropical curves. In: Proceedings of Gökova Geometry: Topology Conference 2006, pp. 39–51. (2007)

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

    Google Scholar 

  31. Nishinou, T., Siebert, B.: Toric degenerations of toric varieties and tropical curves. Duke Math. J. 135(1), 1–51 (2006). arXiv:math/0409060

  32. Okounkov, A., Pandharipande, R.: Gromov-Witten theory, Hurwitz numbers and matrix models. In: I (2001). arXiv:math/0101147

  33. Rau, J.: Tropical intersection theory and gravitational descendants. Ph.D. Thesis (2009)

  34. Severi, F.: Vorlesungen über algebraische Geometrie. Teubner-Verlag, Berlin (1921)

    Book  MATH  Google Scholar 

  35. Speyer, D.: Tropical linear spaces. SIAM J. Discrete Math. 22, 1527–1558 (2008). arXiv:math/0410455

  36. Speyer, D., Sturmfels, B.: The tropical Grassmannian. Adv. Geom. 4(3), 389–411 (2004). arXiv:math/0304218

  37. Vetro, F.: Irreducibility of Hurwitz spaces of coverings with one special fiber and monodromy group a Weyl group of type \(D_d\). Manuscr. Math. 125(3), 353–368 (2008)

    Article  MATH  MathSciNet  Google Scholar 

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Acknowledgments

I would like to thank Hannah Markwig for many inspiring discussions and the anonymous referees for their helpful suggestions. I was supported by DFG Grants MA 4797/3-1 and MA 4797/1-2.

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  1. Institut für Mathematik, Technische Universität Berlin, Sekretariat MA 6-2, Strasse des 17. Juni 136, 10623, Berlin, Germany

    Simon Hampe

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Hampe, S. Combinatorics of tropical Hurwitz cycles. J Algebr Comb 42, 1027–1058 (2015). https://doi.org/10.1007/s10801-015-0615-0

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