# Finite orbits of the pure braid group on the monodromy of the $2$-variable Garnier system

## TL;DR

This paper classifies finite orbits of the braid group action on the monodromy of the 2-variable Garnier system, linking algebraic solutions to exceptional finite orbits on a family of affine varieties.

## Contribution

It provides a classification of exceptional finite orbits of the braid group action on the character variety related to the Garnier system, revealing their geometric structure.

## Key findings

- Character variety of $ ext{SL}_2(C)$ is a 4-dimensional affine family with 5 parameters.
- Braid group action on this family is classified, identifying all finite orbits.
- Finite orbits correspond to algebraic solutions of the Garnier system.

## Abstract

In this paper we show that the $SL_{2}(\mathbb C)$ character variety of the Riemann sphere $\Sigma_5$ with five boundary components is a $5$-parameter family of affine varieties of dimension $4$. We endow this family of affine varieties with an action of the braid group and classify exceptional finite orbits. This action represents the nonlinear monodromy of the $2$ variable Garnier system and finite orbits correspond to algebraic solutions.