Tropical Hurwitz numbers.

*(English)*Zbl 1218.14058Hurwitz numbers are important objects connecting the geometry of algebraic curves to the combinatorics of the symmetric group. Geometrically, they count genus \(g\), degree \(d\) connected covers of the projective line that have a specified ramification profile over a prescribed set of \(n\) points. Via the monodromy representation, this is equivalent to counting \(n\)-tuples of elements of \(S_d\) in prescribed conjugacy classes that multiply to 1 and whose generated subgroup acts transitively on \(1, 2, \dots, d\).

This connection dates back to Hurwitz himself and has provided a rich interplay between the two fields for a long time. In more recent times, Hurwitz numbers have found a prominent role in the study of the moduli spaces of curves and in the Gromov-Witten theory. The moduli space of genus \(g\), degree \(d\) covers of the projective line with only simple ramification (Hurwitz space) admits a natural branch map recording the position of the branch points on the projective line. The degree of the branch map onto its image is tautologically equal to a Hurwitz number. The Hurwitz space sits inside the moduli space of degree \(d\) maps of genus \(g\) stable curves to the projective line. However, the latter is a singular, non-equidimensional stack. B. Fantechi and R. Pandharipande [Compos. Math. 130, No. 3, 345–364 (2002; Zbl 1054.14033)] define a branch map on this stack and show that its virtual degree still recovers the appropriate Hurwitz number.

The theory of relative stable maps [J. Li, J. Differ. Geom. 60, No. 2, 199–293 (2002; Zbl 1063.14069); A.-M. Li and Y. Ruan, Invent. Math. 145, No. 1, 151–218 (2001; Zbl 1062.53073)] extends this scenario to more general Hurwitz numbers, with arbitrary ramification profiles over the branch points.

The paper under review is devoted to the tropical analogue of the above situation in the case of double Hurwitz numbers (i.e., only two branch points can have non-simple ramification profile). In tropical geometry, algebraic curves are replaced by certain piecewise linear objects called tropical curves. The paper develops a tropical counterpart of the branch map (it is defined on an appropriate tropical moduli space which is a weighted polyhedral complex) and shows that its degree recovers classical double Hurwitz numbers. Further, the combinatorial techniques developed are applied to recover results of I. P. Goulden, D. M. Jackson and R. Vakil [Adv. Math. 198, No. 1, 43–92 (2005; Zbl 1086.14022)] and S. Shadrin, M. Shapiro and A. Vainshtein [Adv. Math. 217, No. 1, 79–96 (2008; Zbl 1138.14018)] on the piecewise polynomial structure of double Hurwitz numbers in genus zero.

This connection dates back to Hurwitz himself and has provided a rich interplay between the two fields for a long time. In more recent times, Hurwitz numbers have found a prominent role in the study of the moduli spaces of curves and in the Gromov-Witten theory. The moduli space of genus \(g\), degree \(d\) covers of the projective line with only simple ramification (Hurwitz space) admits a natural branch map recording the position of the branch points on the projective line. The degree of the branch map onto its image is tautologically equal to a Hurwitz number. The Hurwitz space sits inside the moduli space of degree \(d\) maps of genus \(g\) stable curves to the projective line. However, the latter is a singular, non-equidimensional stack. B. Fantechi and R. Pandharipande [Compos. Math. 130, No. 3, 345–364 (2002; Zbl 1054.14033)] define a branch map on this stack and show that its virtual degree still recovers the appropriate Hurwitz number.

The theory of relative stable maps [J. Li, J. Differ. Geom. 60, No. 2, 199–293 (2002; Zbl 1063.14069); A.-M. Li and Y. Ruan, Invent. Math. 145, No. 1, 151–218 (2001; Zbl 1062.53073)] extends this scenario to more general Hurwitz numbers, with arbitrary ramification profiles over the branch points.

The paper under review is devoted to the tropical analogue of the above situation in the case of double Hurwitz numbers (i.e., only two branch points can have non-simple ramification profile). In tropical geometry, algebraic curves are replaced by certain piecewise linear objects called tropical curves. The paper develops a tropical counterpart of the branch map (it is defined on an appropriate tropical moduli space which is a weighted polyhedral complex) and shows that its degree recovers classical double Hurwitz numbers. Further, the combinatorial techniques developed are applied to recover results of I. P. Goulden, D. M. Jackson and R. Vakil [Adv. Math. 198, No. 1, 43–92 (2005; Zbl 1086.14022)] and S. Shadrin, M. Shapiro and A. Vainshtein [Adv. Math. 217, No. 1, 79–96 (2008; Zbl 1138.14018)] on the piecewise polynomial structure of double Hurwitz numbers in genus zero.

Reviewer: P. E. Frenkel (Budapest)

##### MSC:

14T05 | Tropical geometry (MSC2010) |

14H10 | Families, moduli of curves (algebraic) |

##### References:

[1] | Cavalieri, R., Johnson, P., Markwig, H.: Wall crossings for double Hurwitz numbers. In preparation (2009) · Zbl 1231.14023 |

[2] | Ekedahl, T., Lando, S., Shapiro, M., Vainshtein, A.: Hurwitz numbers and intersections on muduli spaces of curves. Invent. Math. 146, 297–327 (2001) · Zbl 1073.14041 · doi:10.1007/s002220100164 |

[3] | Fantechi, B., Pandharipande, R.: Stable maps and branch divisors. Compos. Math. 130(3), 345–364 (2002) · Zbl 1054.14033 · doi:10.1023/A:1014347115536 |

[4] | Gathmann, A., Kerber, M., Markwig, H.: Tropical fans and the moduli space of rational tropical curves. Preprint, arXiv:0708.2268v1 · Zbl 1169.51021 |

[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) · Zbl 0939.05006 · doi:10.1006/jcta.1999.2992 |

[6] | Goulden, I., Jackson, D.M., Vakil, R.: Towards the geometry of double Hurwitz numbers. Adv. Math. 198, 43–92 (2005) · Zbl 1086.14022 · doi:10.1016/j.aim.2005.01.008 |

[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) · Zbl 1088.14007 · doi:10.1215/S0012-7094-05-13011-3 |

[8] | Kerber, M., Markwig, H.: Counting tropical elliptic plane curves with fixed j-invariant. Comment. Math. Helv. 84(2), 387–427 (2009). arXiv:math.AG/0608472 · Zbl 1205.14071 · doi:10.4171/CMH/166 |

[9] | Li, J.: A degeneration formula of GW-invariants. J. Differ. Geom. 60(2), 199–293 (2002) · Zbl 1063.14069 |

[10] | Li, A.-M., Ruan, Y.: Symplectic surgery and Gromov-Witten invariants of Calabi-Yau 3-folds. Invent. Math. 145(1), 151–218 (2001) · Zbl 1062.53073 · doi:10.1007/s002220100146 |

[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). arXiv:math.AG/0601041 · Zbl 1103.14034 |

[12] | Rau, J.: The index of a linear map of lattices. Preprint, TU Kaiserslautern, available at http://www.mathematik.uni-kl.de/\(\sim\)jrau (2006) |

[13] | Shadrin, S., Shapiro, M., Vainshtein, A.: Chamber behavior of double Hurwitz numbers in genus 0. Adv. Math. 217(1), 79–96 (2008) · Zbl 1138.14018 · doi:10.1016/j.aim.2007.06.016 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.