On a Zeuthen-type problem

TL;DR

This paper proves that high-degree meromorphic functions on certain algebraic curves are equivalent to linear projections, and introduces a Zeuthen-type problem related to plane Hurwitz numbers.

Contribution

It establishes a new characterization of meromorphic functions on algebraic curves and formulates a novel Zeuthen-type problem for computing plane Hurwitz numbers.

Findings

Meromorphic functions of degree >4 are linear projections from a point outside the curve.

Introduces a Zeuthen-type problem for calculating plane Hurwitz numbers.

Provides a new perspective on the relationship between algebraic curves and projections.

Abstract

We show that every degree $d$ meromorphic function on a smooth connected projective curve $C\subset \mathbb P^2$ of degree $d>4$ is isomorphic to a linear projection from a point $p\in \mathbb {P}^2 \setminus C$ to $\mathbb P^1$. We then pose a Zeuthen-type problem for calculating the plane Hurwitz numbers.