The moduli space of stable rank 2 parabolic bundles over an elliptic curve with 3 marked points

TL;DR

This paper explicitly describes the moduli space of stable rank 2 parabolic bundles over an elliptic curve with three marked points, revealing its structure as a blow-up of an elliptic curve and linking it to the $SU(2)$ character variety.

Contribution

It provides a detailed geometric description of the moduli space as a blow-up of an elliptic curve and connects it to the $SU(2)$ character variety of the 3-punctured torus.

Findings

Describes the moduli space as a blow-up of an elliptic curve in $( ext{CP}^1)^3$

Reproduces the known Poincaré polynomial for the space

Establishes a link to the $SU(2)$ character variety of the 3-punctured torus

Abstract

We explicitly describe the moduli space $M^s(X,3)$ of stable rank 2 parabolic bundles over an elliptic curve $X$ with trivial determinant bundle and 3 marked points. Specifically, we exhibit $M^s(X,3)$ as a blow-up of an embedded elliptic curve in $(\mathbb{CP}^1)^3$. The moduli space $M^s(X,3)$ can also be interpreted as the $SU(2)$ character variety of the 3-punctured torus. Our description of $M^s(X,3)$ reproduces the known Poincar\'{e} polynomial for this space.