Hyperelliptic odd coverings

TL;DR

This paper classifies hyperelliptic odd coverings of minimal degree, providing explicit counts for general curves using differential equations, monodromy, and deformation techniques.

Contribution

It introduces a comprehensive analysis of hyperelliptic odd coverings, including explicit enumeration and multiple perspectives such as differential equations, monodromy, and deformation.

Findings

Number of hyperelliptic odd coverings of minimal degree is ${3g race g-1} 2^{2g}$ for general curves.

Coverings can be characterized via solutions to specific differential equations.

Monodromy and deformation methods are used to compute the total count.

Abstract

We investigate a class of odd (ramification) coverings $C \to \mathbb{P}^1$ where $C$ is hyperelliptic, its Weierstrass points maps to one fixed point of $\mathbb{P}^1$ and the covering map makes the hyperelliptic involution of $C$ commute with an involution of $\mathbb{P}^1$. We show that the total number of hyperelliptic odd coverings of minimal degree $4g$ is ${3g \choose g-1} 2^{2g}$ when $C$ is general. Our study is approached from three main perspectives: if a fixed effective theta characteristic is fixed they are described as a solution of a certain class of differential equations; then they are studied from the monodromy viewpoint and a deformation argument that leads to the final computation.