Tropical open Hurwitz numbers.

*(English)*Zbl 1226.14066The tropical correspondence theorem of G. Mikhalkin [J. Am. Math. Soc. 18, 313–377 (2005; Zbl 1092.14068)] establishes a (weighted) one-to-one correspondence between the finitely many plane algebraic curves of given genus and degree, passing through suitably many given generic points, and the tropical curves of the same genus and degree, passing through the same number of generic points. Recently, the authors generalized this correspondence to the case where the restrictions imposed on the curves include both passing through given points and tangency to given lines. This setting is classical for enumerative geometry and known as the problem of characteristic numbers of the projective plane.

The weights of tropical curves in this general version of the correspondence theorem turn out to involve a certain generalization of Hurwitz numbers that the authors call open Hurwitz numbers and study in the paper under review. Roughly speaking, an open Hurwitz number is the weighted number of ramified coverings \(f\) over a compact closed oriented surface, such that the behavior of \(f\) is prescribed not only over its critical values, but also over a number of disjoint circles. The authors provide a tropical interpretation of open Hurwitz numbers, generalizing the one obtained for the classical double Hurwitz numbers by P. Cavalieri, P. Johnson and H. Markwig [J. Algebr. Comb. 32, No. 2, 241–265 (2010; Zbl 1218.14058)].

The weights of tropical curves in this general version of the correspondence theorem turn out to involve a certain generalization of Hurwitz numbers that the authors call open Hurwitz numbers and study in the paper under review. Roughly speaking, an open Hurwitz number is the weighted number of ramified coverings \(f\) over a compact closed oriented surface, such that the behavior of \(f\) is prescribed not only over its critical values, but also over a number of disjoint circles. The authors provide a tropical interpretation of open Hurwitz numbers, generalizing the one obtained for the classical double Hurwitz numbers by P. Cavalieri, P. Johnson and H. Markwig [J. Algebr. Comb. 32, No. 2, 241–265 (2010; Zbl 1218.14058)].

Reviewer: Alexander Esterov (Moscow)

##### MSC:

14N10 | Enumerative problems (combinatorial problems) in algebraic geometry |

14P05 | Real algebraic sets |

57M12 | Low-dimensional topology of special (e.g., branched) coverings |

14T05 | Tropical geometry (MSC2010) |

##### References:

[1] | B. Bertrand - E. Brugallé - G. Mikhalkin, Genus 0 characteristic numbers of the tropical projective plane. arXiv:1105.2004. |

[2] | R. Cavalieri - P. Johnson - H. Markwig, Tropical Hurwitz numbers. J. Algebraic Combin. 32, no. 2 (2010), pp. 241-265. Zbl1218.14058 MR2661417 · Zbl 1218.14058 · doi:10.1007/s10801-009-0213-0 |

[3] | S. Lando - A. Zvonkin, Graphs on surfaces and their applications, volume 141 of Encyclopaedia of Mathematical Sciences. Springer-Verlag, Berlin, 2004. With an appendix by Don B. Zagier, Low-Dimensional Topology, II. Zbl1040.05001 MR2036721 · Zbl 1040.05001 |

[4] | T. Nishinou, Disc counting on toric varieties via tropical curves. arXiv:math.AG/0610660. · Zbl 1284.14077 |

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