Towards Plane Hurwitz Numbers

Abstract: The main objects of this thesis are branched coverings obtained as projectionfrom a point in P^2. Our general goal is to understand how a givenmeromorphic function f: X -> P^1 can be induced from a compositionX --> C -> P^1, where C is a plane curve in  P^2 which is birationallyequivalent to the smooth curve X. In particular, we want to characterizemeromorphic functions on plane curves which are obtained in such a way.For instance, we want to describe the relations on branching points ofprojections of plane projective curves of degree d and enumerate such functions.To this end, in a series of two papers, we show that any degree d meromorphicfunction on a smooth projective plane curve C of degree d > 4 isisomorphic to a linear projection from a point p belonging to P^2 \ C to P^1. Secondly,we introduce a planarity filtration of the small Hurwitz space using theminimal degree of a plane curve such that a given meromorphic functioncan be fit into a composition X --> C -> P^1. Finally, we also introduce thenotion of plane Hurwitz numbers in this thesis.