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Counting ramified coverings and intersection theory on Hurwitz spaces. II: Local structure of Hurwitz spaces combinatorial results. (English) Zbl 1131.14037

Let \(S\) denote the Riemann sphere. In this paper, the author investigates the number of ramified coverings \(S\rightarrow S\) with \(c\) prescribed ramification points with ramification types given by fixed partitions \(\kappa_1,\dots,\kappa_c\). Such numbers are called Hurwitz numbers, in honour of A. Hurwitz, who initiated the study of the enumeration of ramified coverings in [Math. Ann. XXXIX. 1–61. (1891; JFM 23.0429.01)]. The main result of the paper under review is the construction of recurrence relations satisfied by Hurwitz numbers.
These relations are obtained by studying the Hurwitz space, a compactification of the space of rational functions of a fixed degree \(n\), and the strata in it containing functions with prescribed multiplicities of critical points and values. The intersection of such a stratum with the border (i.e. the locus of functions defined on nodal curves) of the Hurwitz space is found to be locally isomorphic to a smaller space of smaller dimension. Building on the framework of the first part [S. Lando and D. Zvonkine, Mosc. Math. J. 7, No. 1, 85–107 (2007; Zbl 1131.14034)], this is used to obtain the recurrence relations, which are presented in the form of partial differential equations satisfied by the generating functions of the Hurwitz numbers. As an application, explicit formulas for certain Hurwitz numbers are calculated. Moreover, the generating functions associated to Hurwitz numbers are shown to belong to a subalgebra of the algebra of power series generated by two specific power series.

MSC:

14H30 Coverings of curves, fundamental group
14C17 Intersection theory, characteristic classes, intersection multiplicities in algebraic geometry
05A99 Enumerative combinatorics
14D20 Algebraic moduli problems, moduli of vector bundles
14N10 Enumerative problems (combinatorial problems) in algebraic geometry
14N35 Gromov-Witten invariants, quantum cohomology, Gopakumar-Vafa invariants, Donaldson-Thomas invariants (algebro-geometric aspects)
30F99 Riemann surfaces
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