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Plane Hurwitz Numbers
Abstract
Jared Ongaro
The main objects in this thesis are meromorphic functions obtained
as projections to a pencil of lines through a point in P2 . The general
goal is to understand how a given a meromorphic function f : X → P1
can be induced from a composition X 99K 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.
Division of Pure mathematics
School of Mathematics
University of Nairobi
Kenya
Email:ongaro@uonbi.ac.ke
ISBN: 978-91-7447-927-0
It is shown in this thesis that any degree d > 0 meromorphic function
on a smooth projective 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 99K C → P1 . Additionally, a notion of
plane Hurwitz numbers is introduced.
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