Hostname: page-component-76fb5796d-25wd4 Total loading time: 0 Render date: 2024-04-26T04:07:27.079Z Has data issue: false hasContentIssue false
  • English
  • Français

Moduli of Rank 2 Stable Bundles and Hecke Curves

Published online by Cambridge University Press:  20 November 2018

Sarbeswar Pal
Sarbeswar Pal*
Affiliation:
Indian Institute of Science Education and Research Thiruvananthapuram, Thiruvananthapuram-695016, India e-mail: spal@iisertvm.ac.in
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Let $X$ be a smooth projective curve of arbitrary genus $g\,>\,3$ over the complex numbers. In this short note we will show that the moduli space of rank $2$ stable vector bundles with determinant isomorphic to ${{L}_{x}}$ , where ${{L}_{x}}$ denotes the line bundle corresponding to a point $x\,\in \,X$, is isomorphic to a certain variety of lines in the moduli space of $S$-equivalence classes of semistable bundles of rank $2$ with trivial determinant.

Keywords

14D21Hecke curve(0,1) stable bundle
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 59 , Issue 4 , 01 December 2016 , pp. 865 - 877
Copyright
Copyright © Canadian Mathematical Society 2016

References

[1] Balaji, V., Intermediate facobian of some moduli space of vector bundles on curves. Amer. J. Math. 112(1990), no. 4, 611–629. http://dx.doi.Org/10.2307/2374872 Google Scholar
[2] Cilleruelo, J. and Sols, I., The Severi bound on sections of rank two semistable bundles on a Riemann surface. Ann. ofMath. (2) 154(2001), no. 3, 739–758. http://dx.doi.Org/10.2307/3062146 Google Scholar
[3] Drezet, J.-M. and Narasimhan, M. S., Groupe de Picard des varietes de modules defibres semi-stables sur les courbes algebriques. Invent. Math. 97(1989), no. 1, 53–94. http://dx.doi.Org/10.1007/BF01850655 Google Scholar
[4] Fantechi, B., Göttsche, L., Illusie, L., Kleiman, S., Nitsure, N., and Vistoli, A., Fundamental algebraic geometry. Grothendieck's FGA explained. Mathematical Surveys and Monographs, 123. American Mathematical Society, Providence, RI, 2005.Google Scholar
[5] Hwang, J.-M., Tangent vectors to Hecke curves on the moduli space of rank 2 bundles over an algebraic curve. Duke Math. J. 101(2000), no. 1,179-187. http://dx.doi.Org/10.1215/S0012-7094-00-10117-2 Google Scholar
[6] Hwang, J.-M., Geometry of minimal rational curves on Fano manifolds. In: School on vanishing theorems and effective results in algebraic geometry (Trieste, 2000). ICTP Lect. Notes, 6, Abdus Salam Int. Cent. Theoret. Phys., Trieste, 2001, pp. 335–393,.Google Scholar
[7] Kollár, J., Rational curves on algebraic varieties. Ergebnisse der Mathematik und ihrer Grenzgebiete, 32. Springer-Verlag, Berlin, 1996.Google Scholar
[8] Kilaru, S., Rational curves on moduli spaces of vector bundles. Proc. Indian Acad. Sci. Math. Sci. 108(1998), no. 3, 217–226. http://dx.doi.Org/10.1007/BF02844479 Google Scholar
[9] Mori, S., Projective manifolds with ample tangent bundles. Ann. ofMath. (2) 110(1979), no. 3, 593–606. http://dx.doi.Org/10.2307/1971241 Google Scholar
[10] Mumford, D. and Newstead, P. E., Periods of a moduli space of vector bundles on curves. Amer. J. Math. 90(1968), 1201–1208. http://dx.doi.Org/10.2307/2373296 Google Scholar
[11] Muñoz, V., Quantum cohomology of the moduli space of stable bundles over a Riemann surface. Duke Math. J. 98(1999), no. 3, 525–540. http://dx.doi.Org/10.1215/S0012-7094-99-09816-2 Google Scholar
[12] Muñoz, V., Another proof for the presentation of the quantum cohomology of the moduli of bundles over a Riemann surface. Bull. London Math. Soc. 34(2002), no. 4, 411–414. http://dx.doi.Org/10.1112/S0024609301008906 Google Scholar
[13] Narasimhan, M. S. and Ramanan, S., Deformations of the moduli space of vector bundles over an algebraic curve. Ann. of Math. 101(1975), 391–417. http://dx.doi.Org/10.2307/1 970933 Google Scholar
[14] Narasimhan, M. S. and Ramanan, S., Geometry of Heche Cycles-I. In: C. P. Ramanujam-a tribute. Springer Verlag, Berlin, 1978, pp. 291–345.Google Scholar
[15] Narasimhan, M. S. and Ramanan, S., 29-linear systems on abelian varieties. In: Vector bundles on algebraic varieties (Bombay, 1984), Tata Inst. Fund. Res. Stud. Math., 11. Tata Inst. Fund. Res., Bombay, 1987 pp. 415–427.Google Scholar
[16] Seshadri, C. S., Fibres vectoriels sur les courbes algebriques. Asterisque, 96, Societe Mathematique de France, Paris, 1982.Google Scholar
[17] Xiaotao, S., Minimal rational curves on moduli spaces of stable bundles. Math. Ann. 331(2005), no. 4,925-937. http://dx.doi.Org/10.1007/s00208-004-0614-2 Google Scholar