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Transactions of the American Mathematical Society

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Geometry of the moduli of parabolic bundles on elliptic curves


Author: Néstor Fernández Vargas
Journal: Trans. Amer. Math. Soc. 374 (2021), 3025-3052
MSC (2020): Primary 14H60; Secondary 14D20, 14H52, 14Q10
DOI: https://doi.org/10.1090/tran/7330
Published electronically: February 23, 2021
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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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Additional Information

Néstor Fernández Vargas
Affiliation: IRMAR Université Rennes 1 F-35000 Rennes France
Email: nestor.fernandez-vargas@univ-rennes1.fr

DOI: https://doi.org/10.1090/tran/7330
Keywords: Parabolic vector bundle, parabolic structure, elliptic curve, moduli space, del Pezzo surface
Received by editor(s): November 23, 2016
Received by editor(s) in revised form: June 21, 2017
Published electronically: February 23, 2021
Additional Notes: The author gratefully acknowledges support by the Centre Henri Lebesgue (ANR-11-LABX-0020-01).
Article copyright: © Copyright 2021 American Mathematical Society