Geometry of the moduli of parabolic bundles on elliptic curves

TL;DR

This paper explores the geometric structure of the moduli space of simple rank 2 parabolic bundles on a 2-punctured elliptic curve, revealing its complex gluing structure, embedded curves, and connections to del Pezzo surfaces.

Contribution

It characterizes the moduli space as a non-separated gluing of two charts, proves a Torelli theorem, and links the space to del Pezzo surfaces through elementary transformations.

Findings

Moduli space is a non-separated gluing of two   surfaces.

Embedded curve  isomorphic to the elliptic curve C.

Modular map is a 2:1 cover ramified in .

Abstract

The goal of this paper is the study of simple rank 2 parabolic vector bundles over a $2$-punctured elliptic curve $C$. We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to $\mathbb{P}^1 \times \mathbb{P}^1$. We also showcase a special curve $\Gamma$ isomorphic to $C$ embedded in this space, and this way we prove a Torelli theorem. This moduli space is related to the moduli space of semistable parabolic bundles over $\mathbb{P}^1$ via a modular map which turns out to be the 2:1 cover ramified in $\Gamma$. We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles.