Tropical Hurwitz numbers
Renzo Cavalieri
, Paul Johnson
and Hannah Markwig
DOI: 10.1007/s10801-009-0213-0
Abstract
Hurwitz numbers count genus g, degree d covers of \Bbb P 1 with fixed branch locus. This equals the degree of a natural branch map defined on the Hurwitz space. In tropical geometry, algebraic curves are replaced by certain piece-wise linear objects called tropical curves. This paper develops a tropical counterpart of the branch map and shows that its degree recovers classical Hurwitz numbers. Further, the combinatorial techniques developed are applied to recover results of Goulden et al. (in Adv. Math. 198:43-92, 2005) and Shadrin et al. (in Adv. Math. 217(1):79-96, 2008) on the piecewise polynomial structure of double Hurwitz numbers in genus 0.
Pages: 241–265
Keywords: Hurwitz numbers; tropical curves
Full Text: PDF
References
1. Cavalieri, R., Johnson, P., Markwig, H.: Wall crossings for double Hurwitz numbers. In preparation (2009)
2. Ekedahl, T., Lando, S., Shapiro, M., Vainshtein, A.: Hurwitz numbers and intersections on muduli spaces of curves. Invent. Math. 146, 297-327 (2001)
3. Fantechi, B., Pandharipande, R.: Stable maps and branch divisors. Compos. Math. 130(3), 345-364 (2002)
4. Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli space of rational tropical curves. Preprint,
5. Goulden, I., Jackson, D.: A proof of a conjecture for the number of ramified covers of the sphere by the torus. J. Comb. Theory 88(2), 246-258 (1999)
6. Goulden, I., Jackson, D.M., Vakil, R.: Towards the geometry of double Hurwitz numbers. Adv. Math. 198, 43-92 (2005)
7. 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)
8. Kerber, M., Markwig, H.: Counting tropical elliptic plane curves with fixed j -invariant. Comment. Math. Helv. 84(2), 387-427 (2009).
9. Li, J.: A degeneration formula of GW-invariants. J. Differ. Geom. 60(2), 199-293 (2002)
10. Li, A.-M., Ruan, Y.: Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds. Invent. Math. 145(1), 151-218 (2001)
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12. Rau, J.: The index of a linear map of lattices. Preprint, TU Kaiserslautern, available at (2006)
13. Shadrin, S., Shapiro, M., Vainshtein, A.: Chamber behavior of double Hurwitz numbers in genus 0.
2. Ekedahl, T., Lando, S., Shapiro, M., Vainshtein, A.: Hurwitz numbers and intersections on muduli spaces of curves. Invent. Math. 146, 297-327 (2001)
3. Fantechi, B., Pandharipande, R.: Stable maps and branch divisors. Compos. Math. 130(3), 345-364 (2002)
4. Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli space of rational tropical curves. Preprint,
5. Goulden, I., Jackson, D.: A proof of a conjecture for the number of ramified covers of the sphere by the torus. J. Comb. Theory 88(2), 246-258 (1999)
6. Goulden, I., Jackson, D.M., Vakil, R.: Towards the geometry of double Hurwitz numbers. Adv. Math. 198, 43-92 (2005)
7. 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)
8. Kerber, M., Markwig, H.: Counting tropical elliptic plane curves with fixed j -invariant. Comment. Math. Helv. 84(2), 387-427 (2009).
9. Li, J.: A degeneration formula of GW-invariants. J. Differ. Geom. 60(2), 199-293 (2002)
10. Li, A.-M., Ruan, Y.: Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds. Invent. Math. 145(1), 151-218 (2001)
11. Mikhalkin, G.: Tropical geometry and its applications. In: Sanz-Sole, M. et al. (eds.) Invited Lectures, vol. II. Proceedings of the ICM Madrid, pp. 827-852 (2006).
12. Rau, J.: The index of a linear map of lattices. Preprint, TU Kaiserslautern, available at (2006)
13. Shadrin, S., Shapiro, M., Vainshtein, A.: Chamber behavior of double Hurwitz numbers in genus 0.
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