Plane Hurwitz Number

Abstract

The main objects in this thesis are meromorphic functions obtained as projections to a p encil of lines through a p oi nt in P2. The general goal is to understand how a given a meromorphic function f ∶ X → P1 can be induced from a composition X ⇢ C → P1, where C ⊂ P2 is birationally equivalent to the smooth curve X. In particular, it is the desire to characterize meromorphic functions on smooth curves which are obtained in such a way and enumerate such functions. It is shown in this thesis that any degree d meromorphic function on a smooth projective plane curve C ⊂ P2 of degree d > 4 is isomorphic to a linear projection from a point p ∈ P2/C to P1. Further, a planarity filtration of the small Hurwitz space using the minimal degree of a plane curve is introduced such that a given meromorphic function admits such a composition X ⇢ C → P1. Additionally, a notion of plane Hurwitz numbers is introduced.