## Geometry of the moduli of parabolic bundles on elliptic curves

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- by Néstor Fernández Vargas PDF
- 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.## References

- M. F. Atiyah,
*Vector bundles over an elliptic curve*, Proc. London Math. Soc. (3)**7**(1957), 414–452. MR**131423**, DOI 10.1112/plms/s3-7.1.414 - V. Balaji, Indranil Biswas, and Sebastian del Baño Rollin,
*A Torelli type theorem for the moduli space of parabolic vector bundles over curves*, Math. Proc. Cambridge Philos. Soc.**130**(2001), no. 2, 269–280. MR**1806777**, DOI 10.1017/S0305004100004916 - Stefan Bauer,
*Parabolic bundles, elliptic surfaces and $\textrm {SU}(2)$-representation spaces of genus zero Fuchsian groups*, Math. Ann.**290**(1991), no. 3, 509–526. MR**1116235**, DOI 10.1007/BF01459257 - Indranil Biswas, Yogish I. Holla, and Chanchal Kumar,
*On moduli spaces of parabolic vector bundles of rank 2 over $\Bbb C\Bbb P^1$*, Michigan Math. J.**59**(2010), no. 2, 467–475. MR**2677632**, DOI 10.1307/mmj/1281531467 - Hans U. Boden and Yi Hu,
*Variations of moduli of parabolic bundles*, Math. Ann.**301**(1995), no. 3, 539–559. MR**1324526**, DOI 10.1007/BF01446645 - Michele Bolognesi,
*On Weddle surfaces and their moduli*, Adv. Geom.**7**(2007), no. 1, 113–144. MR**2290643**, DOI 10.1515/ADVGEOM.2007.008 - Michele Bolognesi,
*A conic bundle degenerating on the Kummer surface*, Math. Z.**261**(2009), no. 1, 149–168. MR**2452642**, DOI 10.1007/s00209-008-0319-4 - C. Casagrande,
*Rank $2$ quasiparabolic vector bundles on $\Bbb {P}^1$ and the variety of linear subspaces contained in two odd-dimensional quadrics*, Math. Z.**280**(2015), no. 3-4, 981–988. MR**3369361**, DOI 10.1007/s00209-015-1458-z - Igor V. Dolgachev,
*Classical algebraic geometry*, Cambridge University Press, Cambridge, 2012. A modern view. MR**2964027**, DOI 10.1017/CBO9781139084437 - Hélène Esnault and Claus Hertling,
*Semistable bundles on curves and reducible representations of the fundamental group*, Internat. J. Math.**12**(2001), no. 7, 847–855. MR**1850674**, DOI 10.1142/S0129167X01001003 - V. Heu and F. Loray,
*Flat rank $2$ vector bundles on genus $2$ curves*, Mem. Amer. Math. Soc. (to appear), January 2014. - Chanchal Kumar,
*Invariant vector bundles of rank 2 on hyperelliptic curves*, Michigan Math. J.**47**(2000), no. 3, 575–584. MR**1813545**, DOI 10.1307/mmj/1030132595 - Yves Laszlo,
*About $G$-bundles over elliptic curves*, Ann. Inst. Fourier (Grenoble)**48**(1998), no. 2, 413–424. MR**1625614**, DOI 10.5802/aif.1623 - F. Loreĭ,
*Isomonodromic deformations of Lamé connections, the Painlevé VI equation and Okamoto symmetry*, Izv. Ross. Akad. Nauk Ser. Mat.**80**(2016), no. 1, 119–176 (Russian, with Russian summary); English transl., Izv. Math.**80**(2016), no. 1, 113–166. MR**3462678**, DOI 10.4213/im8310 - Frank Loray and Masa-Hiko Saito,
*Lagrangian fibrations in duality on moduli spaces of rank 2 logarithmic connections over the projective line*, Int. Math. Res. Not. IMRN**4**(2015), 995–1043. MR**3340345**, DOI 10.1093/imrn/rnt232 - Francois-Xavier Machu,
*Monodromy of a class of logarithmic connections on an elliptic curve*, SIGMA Symmetry Integrability Geom. Methods Appl.**3**(2007), Paper 082, 31. MR**2366940**, DOI 10.3842/SIGMA.2007.082 - Masaki Maruyama,
*On automorphism groups of ruled surfaces*, J. Math. Kyoto Univ.**11**(1971), 89–112. MR**280493**, DOI 10.1215/kjm/1250523688 - M. Maruyama and K. Yokogawa,
*Moduli of parabolic stable sheaves*, Math. Ann.**293**(1992), no. 1, 77–99. MR**1162674**, DOI 10.1007/BF01444704 - V. B. Mehta and C. S. Seshadri,
*Moduli of vector bundles on curves with parabolic structures*, Math. Ann.**248**(1980), no. 3, 205–239. MR**575939**, DOI 10.1007/BF01420526 - C. S. Seshadri,
*Moduli of vector bundles on curves with parabolic structures*, Bull. Amer. Math. Soc.**83**(1977), no. 1, 124–126. MR**570987**, DOI 10.1090/S0002-9904-1977-14210-9 - Loring W. Tu,
*Semistable bundles over an elliptic curve*, Adv. Math.**98**(1993), no. 1, 1–26. MR**1212625**, DOI 10.1006/aima.1993.1011

## Additional Information

**Néstor Fernández Vargas**- Affiliation: IRMAR, Université Rennes 1, F-35000 Rennes, France
- Email: nestor.fernandez-vargas@univ-rennes1.fr
- 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).
- © Copyright 2021 American Mathematical Society
- 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
- MathSciNet review: 4237941