The problem of studying the intersection theory on Hurwitz spaces
naturally arises in connection with the Hurwitz problem. The latter
consists in enumeration of ramified coverings of the $2$-sphere,
with prescribed ramification data. It is also closely related
to the computation of Gromov--Witten invariants --- the invariants
of compact complex manifolds counting maps of complex curves
to the manifold. The study of the intersection theory on Hurwitz
spaces has been initiated in [1,4].
We discuss a new approach to the intersection theory on Hurwitz
spaces. This approach is based on the theory of universal
polynomials originating in the work by R. Thom in early 60ies.
It concerns general holomorphic mappings $f:M\to N$
of compact complex manifolds (which we assume, for definiteness,
being of the same dimension). The main theorems of the theory postulate,
in various situations, the
existence of universal polynomials in the relative Chern classes of $f$,
expressing the cohomology
classes of the closures of loci of points in $M$
where $f$ acquires given singularities.
This theory was recently extended
to the case of multisingularities by M. Kazarian [2]
whose results allow one to describe as universal
polynomials in the pushforwards of the relative Chern classes
the cohomology classes in $N$ represented by the loci of
points with prescribed singularities at the preimages.
We apply the theories of universal polynomials
(and their further extensions)
to the Hurwitz spaces treated as families of meromorphic
functions on complex curves [3]. Since the domain of definition
of a function is one dimensional, the classification of
singularities is relatively simple, and there is a hope
for a complete description of all necessary cohomology
classes. Up to now we have obtained only partial results
expressing the desired cohomology classes in the Hurwitz spaces
in terms of so-called ``basic'' classes related to the
singularities and relative Chern classes.
However, already these partial results led
to previously unknown enumeration formulas.
We hope that our approach will find a much wider domain of
application and consider the Hurwitz problem only as
an important example where necessary tools
useful in general situation can be developed.
[1] T. Ekedahl, S. K. Lando, M. Shapiro, A. Vainshtein
{\it Hurwitz numbers and intersections on moduli spaces of curves},
Invent. Math. {\bf 146} (2001), no. 2, 297--327.
[2] M. E. Kazarian
{\it Multisingularities, cobordisms, and enumerative geometry},
Russ. Math. Surv. {\bf 58} (2003), no. 4, 665--724.
[3] M. E. Kazarian, S. K. Lando,
{\it On intersection theory on Hurwitz spaces},
Izv. Ross. Akad. Nauk Ser. Mat. {\bf 68} (2004), no. 5, 91-122
[4] S. K. Lando, D. Zvonkine,
{\it Counting ramified coverings and intersection
theory on spaces of rational functions I (Cohomology of Hurwitz spaces)},
preprint MPI 2003 - 48, Bonn