Talk at Arnold's seminar
March 6th, 2001

Maxim Kazarian

Goryunov-Lando Compactification of Moduli Spaces and Hurwitz Numbers

The problem of topological classification of ramified coverings 
over a sphere was studied first by Hurwitz. Arnold applied the
quasihomogeneity of the Lyashko-Loijenga map and gave a topological
proof of this combinatorial problem.

Later Arnold's approach was extended to some variations of Hurwitz's
problem in several papers, in particular, due to Goryunov-Lando and
Ekedahl-Lando-Vainshtein-Shapiro. The relation between their
constructions was not clarified in those papers. For example, the
Hurwitz numbers in [ELVS] is expressed in terms of certain integral
over the moduli space of marked curves and it was not clear whether
the paper [GL] contains implicitly computation of this integral (in
case of genus 0).

In this talk we explain the relation between the two constructions:
they are really very close. It happens that the paper [GL] has some
gap and filling this gap leads to the same integral over the moduli
space. So the computation of this integral cannot be avoided (its 
computation is not difficult in case of genus 0 though).

The subject of this talk has appeared after many interesting
discussions with V. Goryunov, M. Shapiro, S. Lando.