A Torelli theorem for moduli spaces of parabolic vector bundles over an elliptic curve

Fassarella, Thiago; Justo, Luana

doi:10.1090/proc/15937

A Torelli theorem for moduli spaces of parabolic vector bundles over an elliptic curve
HTML articles powered by AMS MathViewer

by Thiago Fassarella and Luana Justo PDF

Proc. Amer. Math. Soc. 150 (2022), 1849-1863 Request permission

Abstract:

Let $C$ be an elliptic curve, $w\in C$, and let $S\subset C$ be a finite subset of cardinality at least $3$. We prove a Torelli type theorem for the moduli space of rank two parabolic vector bundles with determinant line bundle $\mathcal O_C(w)$ over $(C,S)$ which are semistable with respect to a weight vector $\big (\frac {1}{2}, \dots , \frac {1}{2}\big )$.

References

Carolina Araujo, Thiago Fassarella, Inder Kaur, and Alex Massarenti, On automorphisms of moduli spaces of parabolic vector bundles, Int. Math. Res. Not. IMRN 3 (2021), 2261–2283. MR 4206612, DOI 10.1093/imrn/rnz132
David Alfaya and Tomás L. Gómez, Automorphism group of the moduli space of parabolic bundles over a curve, Adv. Math. 393 (2021), Paper No. 108070, 127. MR 4339527, DOI 10.1016/j.aim.2021.108070
Carolina Araujo and Alex Massarenti, Explicit log Fano structures on blow-ups of projective spaces, Proc. Lond. Math. Soc. (3) 113 (2016), no. 4, 445–473. MR 3556488, DOI 10.1112/plms/pdw034
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
Indranil Biswas, Tomás L. Gómez, and Vicente Muñoz, Automorphisms of moduli spaces of vector bundles over a curve, Expo. Math. 31 (2013), no. 1, 73–86. MR 3035121, DOI 10.1016/j.exmath.2012.08.002
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
Usha Bhosle, Generalised parabolic bundles and applications to torsionfree sheaves on nodal curves, Ark. Mat. 30 (1992), no. 2, 187–215. MR 1289750, DOI 10.1007/BF02384869
Usha N. Bhosle, Generalized parabolic bundles and applications. II, Proc. Indian Acad. Sci. Math. Sci. 106 (1996), no. 4, 403–420. MR 1425615, DOI 10.1007/BF02837696
Arnaud Beauville, M. S. Narasimhan, and S. Ramanan, Spectral curves and the generalised theta divisor, J. Reine Angew. Math. 398 (1989), 169–179. MR 998478, DOI 10.1515/crll.1989.398.169
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
R. Hartshorne, Algebraic geometry, Springer-Verlag, 1997.
Jun-Muk Hwang and S. Ramanan, Hecke curves and Hitchin discriminant, Ann. Sci. École Norm. Sup. (4) 37 (2004), no. 5, 801–817 (English, with English and French summaries). MR 2103475, DOI 10.1016/j.ansens.2004.07.001
Alexis Kouvidakis and Tony Pantev, The automorphism group of the moduli space of semistable vector bundles, Math. Ann. 302 (1995), no. 2, 225–268. MR 1336336, DOI 10.1007/BF01444495
Marina Logares and Johan Martens, Moduli of parabolic Higgs bundles and Atiyah algebroids, J. Reine Angew. Math. 649 (2010), 89–116. MR 2746468, DOI 10.1515/CRELLE.2010.090
Masaki Maruyama, Elementary transformations in the theory of algebraic vector bundles, Algebraic geometry (La Rábida, 1981) Lecture Notes in Math., vol. 961, Springer, Berlin, 1982, pp. 241–266. MR 708337, DOI 10.1007/BFb0071286
D. Mumford and P. Newstead, Periods of a moduli space of bundles on curves, Amer. J. Math. 90 (1968), 1200–1208. MR 234958, DOI 10.2307/2373296
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
S. Mukai, Finite generation of the nagata invariant rings in A-D-E cases, RIMS Preprint, 1502, 2005.
M. S. Narasimhan and S. Ramanan, Deformations of the moduli space of vector bundles over an algebraic curve, Ann. of Math. (2) 101 (1975), 391–417. MR 384797, DOI 10.2307/1970933
M. S. Narasimhan and C. S. Seshadri, Stable and unitary vector bundles on a compact Riemann surface, Ann. of Math. (2) 82 (1965), 540–567. MR 184252, DOI 10.2307/1970710
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
A. N. Tjurin, Analogues of Torelli’s theorem for multidimensional vector bundles over an arbitrary algebraic curve, Izv. Akad. Nauk SSSR Ser. Mat. 34 (1970), 338–365 (Russian). MR 0265372
Néstor Fernández Vargas, Geometry of the moduli of parabolic bundles on elliptic curves, Trans. Amer. Math. Soc. 374 (2021), no. 5, 3025–3052. MR 4237941, DOI 10.1090/tran/7330
Kôji Yokogawa, Compactification of moduli of parabolic sheaves and moduli of parabolic Higgs sheaves, J. Math. Kyoto Univ. 33 (1993), no. 2, 451–504. MR 1231753, DOI 10.1215/kjm/1250519269
Kôji Yokogawa, Infinitesimal deformation of parabolic Higgs sheaves, Internat. J. Math. 6 (1995), no. 1, 125–148. MR 1307307, DOI 10.1142/S0129167X95000092