# Counting surface branched covers

## TL;DR

This paper investigates the enumeration of branched covers of surfaces with given branch data, revealing only three distinct counts under various natural equivalence relations using Grothendieck's dessins d'enfant.

## Contribution

It classifies the number of distinct branched covers for a given branch datum, showing only three possible counts under different natural restrictions.

## Key findings

- Number of distinct branched covers is always three under various conditions.
- Uses Grothendieck's dessins d'enfant to distinguish the counts.
- Provides a detailed analysis of the Hurwitz problem and its refinements.

## Abstract

To a branched cover f between orientable surfaces one can associate a certain branch datum D(f), that encodes the combinatorics of the cover. This D(f) satisfies a compatibility condition called the Riemann-Hurwitz relation. The old but still partly unsolved Hurwitz problem asks whether for a given abstract compatible branch datum D there exists a branched cover f such that D(f)=D. One can actually refine this problem and ask how many these f's exist, but one must of course decide what restrictions one puts on such f's, and choose an equivalence relation up to which one regards them. And it turns out that quite a few natural choices are possible. In this short note we carefully analyze all these choices and show that the number of actually distinct ones is only three. To see that these three choices are indeed different we employ Grothendieck's dessins d'enfant.