# Realizations of certain odd-degree surface branch data

## TL;DR

This paper investigates the realizability of certain odd-degree surface branch data with three branch points, providing explicit counts of realizations for small cases and linking these to arithmetic properties and conjectures.

## Contribution

It explicitly computes the number of realizations for specific small cases and relates realizability to arithmetic properties of the partitions, supporting existing conjectures.

## Key findings

- Explicit counts of realizations for small h and L values.
- Realizability depends on arithmetic properties of partition entries.
- Confirmed cases where realizability is obstructed by common divisors.

## Abstract

We consider surface branch data with base surface the sphere, odd degree d, three branching points, and two partitions of d of the form (2,...,2,1) and (2,...,2,2h+1). If the third partition has length L, this datum satisfies the Riemann-Hurwitz necessary condition for realizability if h-L is odd and at least -1. For several small values of h and L (namely, for h+L<6) we explicitly compute the number n of realizations of the datum up to the equivalence relation given by the action of automorphisms (even unoriented ones) of both the base and the covering surface. The expression of n depends on arithmetic properties of the entries of the third partition. In particular we find that in the only case where n is 0 these entries have a common divisor, in agreement with a conjecture of Edmonds-Kulkarny-Stong and a stronger one of Zieve.