Artin braid groups and spin structures

TL;DR

This paper explores how the Artin braid group acts on spin structures of hyperelliptic curves, providing a combinatorial method to classify orbits and isotropy groups, advancing understanding in algebraic geometry and group actions.

Contribution

It offers a new combinatorial approach to compute orbits and isotropy groups of spin structures under braid group actions, extending previous geometric descriptions.

Findings

Computed S_{2g+2}-orbits of spin structures

Determined isotropy groups for each orbit

Provided a purely combinatorial framework

Abstract

We study the action of the Artin braid group B_{2g+2} on the set of spin structures on a hyperelliptic curve of genus g, which reduces to that of the symmetric group. It has been already described in terms of the classical theory of Riemann surfaces. In this paper, we compute the S_{2g+2}-orbits of the spin structures of genus $g$ and the isotropy group G_i of each orbit in a purely combinatorial way.