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Transactions of the American Mathematical Society

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Verlinde bundles and generalized theta linear series


Author: Mihnea Popa
Journal: Trans. Amer. Math. Soc. 354 (2002), 1869-1898
MSC (2000): Primary 14H60; Secondary 14J60
DOI: https://doi.org/10.1090/S0002-9947-01-02923-3
Published electronically: November 5, 2001
MathSciNet review: 1881021
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Abstract | References | Similar Articles | Additional Information

Abstract: In this paper we approach the study of generalized theta linear series on moduli of vector bundles on curves via vector bundle techniques on abelian varieties.

We study a naturally defined class of vector bundles on a Jacobian, called Verlinde bundles, in order to obtain information about duality between theta functions and effective global and normal generation on these moduli spaces.


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  • 1. A. Beauville, Vector bundles on curves and generalized theta functions: recent results and open problems, in Current topics in algebraic geometry, Cambridge Univ. Press (1995), 17-33 MR 97h:14015
  • 2. A. Beauville, M.S. Narasimhan and S. Ramanan, Spectral curves and the generalized theta divisor, J. Reine Angew. Math. 398 (1989), 169-179 MR 91c:14040
  • 3. S. Brivio and A. Verra, The theta divisor of $SU_{C}(2)^{s}$is very ample if $C$ is not hyperelliptic, Duke Math. J. 82 (1996), 503-552 MR 97e:14070
  • 4. R. Donagi and L. Tu, Theta functions for SL(n) versus GL(n), Math. Res. Let. 1 (1994), 345-357 MR 95j:14012
  • 5. J.-M. Drezet and M.S. Narasimhan, Groupe de Picard des variétés des modules de fibrés semi-stables sur les courbes algébriques, Invent. Math. 97 (1989), 53-94 MR 90d:14008
  • 6. B. van Geemen and E. Izadi, The tangent space to the moduli space of vector bundles on a curve and the singular locus of the theta divisor of the Jacobian, Journal of Alg. Geom. 10 (2001), 133-177 CMP 2001:04
  • 7. P.A. Griffiths and J. Harris, Principles of algebraic geometry, Wiley-Interscience 1978 MR 80b:14001
  • 8. R. Hartshorne, Algebraic geometry, Springer-Verlag 1977 MR 57:3116
  • 9. G. Hein, On the generalized theta divisor, Contrib. to Alg. and Geom. 38 (1997), No.1, 95-98 MR 98a:14018
  • 10. A. Hirschowitz, Problemes de Brill-Noether en rang superieur, C. R. Acad. Sci. Paris Sér. I Math. 307 (1988), 153-156 MR 89i:14010
  • 11. D. Huybrechts and M. Lehn, The geometry of moduli spaces of sheaves, Aspects of Mathematics, Vieweg 1997 MR 98g:14012
  • 12. G. Kempf, Projective coordinate rings of abelian varieties, in Algebraic analysis, geometry and number theory, J.I.Igusa ed., Johns Hopkins Press (1989), 225-236 MR 98m:14047
  • 13. S. Koizumi, Theta relations and projective normality of abelian varieties, Am. J. Math. 98 (1976), 865-889 MR 58:702
  • 14. Y. Laszlo, A propos de l'espace des modules des fibrés de rang 2 sur une courbe, Math. Ann. 299 (1994), 597-608 MR 95f:14021
  • 15. R. Lazarsfeld, A sampling of vector bundle techniques in the study of linear series, in Lectures on Riemann surfaces, World Scientific (1989), 500-559 MR 92f:14006
  • 16. J. Le Potier, Module des fibrés semi-stables et fonctions thêta, Proc. Symp. Taniguchi Kyoto 1994: Moduli of vector bundles, M. Maruyama ed., Lecture Notes in Pure and Appl. Math. 179 (1996), 83-101 MR 97f:14013
  • 17. S. Mukai, Duality between $D(X)$ and $D(\widehat{X})$ with its application to Picard sheaves, Nagoya Math. J. 81 (1981), 153-175 MR 82f:14036
  • 18. S. Mukai, Fourier functor and its application to the moduli of bundles on an abelian variety, In: Algebraic Geometry, Sendai 1985, Advanced studies in pure mathematics 10 (1987), 515-550 MR 89k:14026
  • 19. D. Mumford, On the equations defining abelian varieties, Invent. Math. 1 (1966), 287-354 MR 34:4269
  • 20. D. Mumford, Abelian varieties, Second edition, Oxford Univ. Press 1974 MR 44:219
  • 21. M.S. Narasimhan and S. Ramanan, Generalized Prym varieties as fixed points, J. Indian Math. Soc., 39 (1975), 1-19 MR 54:12777
  • 22. G. Pareschi, Syzygies of abelian varieties, J. Amer. Math. Soc. 13 (2000), 651-654 MR 2001f:14086
  • 23. M. Popa, On the base locus of the generalized theta divisor, C. R. Acad. Sci. Paris Série I, 329 (1999), 507-512 MR 2000j:14052
  • 24. M. Popa, Dimension estimates for Hilbert schemes and effective base point freeness on moduli spaces of vector bundles on curves, Duke Math. J. 107 (2001), 469-495.
  • 25. M. Raynaud, Sections des fibrés vectoriels sur une courbe, Bull. Soc. Math. France 110 (1982), 103-125. MR 84a:14009
  • 26. T. Sekiguchi, On the normal generation by a line bundle on an abelian variety, Proc. Japan Acad. 54 (1978), 185-188 MR 80c:13026

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Additional Information

Mihnea Popa
Affiliation: Department of Mathematics, University of Michigan, 525 East University, Ann Arbor, Michigan 48109-1109; Institute of Mathematics of the Romanian Academy, Bucharest, Romania
Address at time of publication: Department of Mathematics, Harvard University, Cambridge, Massachusetts 02138
Email: mpopa@math.harvard.edu

DOI: https://doi.org/10.1090/S0002-9947-01-02923-3
Keywords: Vector bundles, nonabelian theta functions
Received by editor(s): March 1, 2001
Published electronically: November 5, 2001
Article copyright: © Copyright 2001 American Mathematical Society

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