Geometry of the moduli of parabolic bundles on elliptic curves

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$\mathrm{.}Wealsoshowcaseaspecialcurve$\Gamma$isomorphicto$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 $\mathrm{\Gamma }$. We recover the geometry of del Pezzo surfaces of degree 4 and we reconstruct all their automorphisms via elementary transformations of parabolic vector bundles.