Geometry of the moduli of parabolic bundles on elliptic curves

Abstract

The goal of this paper is the study of simple rank 2 parabolic vector bundles over a 2 2 -punctured elliptic curve C C . We show that the moduli space of these bundles is a non-separated gluing of two charts isomorphic to P 1 × P 1 \mathbb {P}^1 \times \mathbb {P}^1 . We also showcase a special curve Γ \Gamma isomorphic to C 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 P 1 \mathbb {P}^1 via a modular map which turns out to be the 2:1 cover ramified in Γ \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.