Towards Plane Hurwitz Numbers

Abstract

The main objects of this thesis are branched coverings obtained as projection from a point in P^2. Our general goal is to understand how a given meromorphic function f: X -> P^1 can be induced from a composition X --> C -> P^1, where C is a plane curve in P^2 which is birationally equivalent to the smooth curve X. In particular, we want to characterize meromorphic functions on plane curves which are obtained in such a way. For instance, we want to describe the relations on branching points of projections 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 meromorphic function on a smooth projective plane curve C of degree d > 4 is isomorphic 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 the minimal degree of a plane curve such that a given meromorphic function can be fit into a composition X --> C -> P^1. Finally, we also introduce the notion of plane Hurwitz numbers in this thesis.