Ramified coverings of the two-dimensional sphere and the intersection theory in spaces

S. K. Lando

Trudy Matematicheskogo Instituta imeni V.A. Steklova, Tome 57 (2002) no. 3, pp. 463-533

Résumé

In 1891 A. Hurwitz considered the problem of enumerating the ramified coverings of the two-dimensional sphere by two-dimensional surfaces with fixed types of branching over the branch points. In the original setting the problem was reformulated in terms of characters of the symmetric group. Recently it turned out that the problem is also very closely connected with diverse physical theories, with singularity theory, and with the geometry of the moduli spaces of complex curves. The discovery of these relationships has led to an enlargement of the class of cases in which the enumeration yields explicit formulae, and a clarification of the nature of the classical results. This survey is devoted to a description of the contemporary state of this thriving topic and is intended for experts in topology, the theory of Riemann surfaces, combinatorics, singularity theory, and mathematical physics. It can also serve as a guide to the modern literature on coverings of the sphere.