On the Hurwitz existence problem for branched covers of the projective line

TL;DR

This paper provides an alternative proof for the Hurwitz existence problem concerning branched covers of the projective line, specifically when ramification profiles are of a simple form with a single nontrivial part.

Contribution

It offers a new proof approach for a special case of the Hurwitz existence problem involving specific ramification profiles.

Findings

Established an alternative proof for the Hurwitz existence problem in the specified case.

Clarified conditions under which branched covers with certain ramification profiles exist.

Abstract

We give an alternative proof of the Hurwitz existence problem for branched covers of $\mathbb{P}^1$ in the case where the number of ramification points equals the number of branch points, that is, where all the ramification profiles are of the form $[e,1,\ldots,1]$ with $e \geq 2$.