Geometry of the moduli of parabolic bundles on elliptic curves

Fernández Vargas, Néstor

doi:10.1090/tran/7330

Geometry of the moduli of parabolic bundles on elliptic curves
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Trans. Amer. Math. Soc. 374 (2021), 3025-3052 Request permission

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.

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