Jared Ongaro: Towards Plane Hurwitz Numbers

The main objects in this thesis are meromorphic functions obtained as projections to a pencil of lines through a point in P². Our general goal is to understand how a given meromorphic function f: X → P¹ can be induced from a composition X → C → P¹, where C is a plane curve birationally equivalent to the smooth curve X. In particular, we want to characterize meromorphic functions on smooth curves which are obtained in such a way and enumerate such functions.

In a series of two papers, we first 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 in P². 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 admits such a composition X → C → P¹. Additionally, a notion of plane Hurwitz numbers is introduced in this thesis.