Vector Bundles on Singular Projective Curves

  • I. Burban
  • Yu. Drozd
  • G.-M Greuel
Chapter
Part of the NATO Science Series book series (NAII, volume 36)

Abstract

In this survey artiele we report on reeent results known for vector bundles on singular projective curves (see (Drozd and Greuel; Drozd, Greuel and Kashuba; Yudin). We recall the description of vector bundles on tame and finite configurations of projective lines using the combinatorics of matrix problems. We also show that this combinatorics allows us to compute the cohomology groups of a vector bundle, the dual bundle of a vector bundle, the tensor product of two vector bundles, the dimension of the homomorphism spaces between two vector bundles, and finally to classify simple vector bundles.

Key words

tame and wild representation type matrix problems vector bundles on curves 

Mathematics Subject Classification (2000)

14H60 16G60 16G50 

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References

  1. Atiyah, M. (1957) Vector bundles over an elliptic curve, Proc. London Math. Soc. 7, 414–452.MathSciNetMATHCrossRefGoogle Scholar
  2. Bass, H. (1963) On the ubiquity of Gorenstein rings, Math. Zeitsch. 82, 8–27.MathSciNetMATHCrossRefGoogle Scholar
  3. Bondarenko, V. V. (1988) Bundles of semi-chains and their representations, preprint of the Kiev Institute of mathematics.Google Scholar
  4. Bondarenko, V. V. (1992) Representations of bundles of semi-chains and their applications, St. Petersburg Math. J. 3, 973–996.MathSciNetGoogle Scholar
  5. Bondarenko, V. V., Nazarova, L. A., Roiter, A. V., and Sergijchuck, V. V. (1972), Applications of the modules over a diad to the classification of finite p-groups, having an abelian subgroup of index p, Zapiski Nauchn. Seminara LOMI 28, 69–92.Google Scholar
  6. Dro/d, Yu. A. (1972) Matrix problems and categories of matrices, Zapiski Nauchn. Seminara LOMI 28, 144–153.Google Scholar
  7. Drozd, Yu. A., and Greuel, G.-M. (1999) On the classification of vector bundles on projective curves, Max-Plank-Institut für Mathematik Preprint Series 130.Google Scholar
  8. Drozd, Yu. A., Greuel, G.-M., and Kashuba, I. M. (2000) On Cohen-Macaulay Modules on Surface Singularities, preprint, Max-Planck-Institut für Mathematik Bonn.Google Scholar
  9. Gelfand, I. M. (1970) Cohomology of the infinite dimensional Lie algebras; some questions of the integral geometry, International congress of mathematics, Nice.Google Scholar
  10. Gelfand, I. M., and Ponomarcv, V. A. (1968) Indecomposable representations of the Lorenz group, Uspehi Mat. Nauk 140, 3–60.Google Scholar
  11. Grothendieck, A. (1956) Sur la classification des fibres holomorphes sur la sphère de Riemann, Amer. J. Math. 79, 121–138.MathSciNetCrossRefGoogle Scholar
  12. Hartshorne, R. (1977) Algebraic Geometry, Springer.Google Scholar
  13. Nazarova, L. A., and Roiter, A. V. (1969) Finitely generated modules over diad of two discrete valuation rings, Izv. Akad. Nauk USSR 33, 65–89.MathSciNetMATHCrossRefGoogle Scholar
  14. Nazarova, L. A., and Roiter, A. V. (1973) About one problem of 1. M. Gelfand, Functional analysis and its applications 4, 54–69.MathSciNetGoogle Scholar
  15. Polishchuk, A. (2000) Classical Yang-Baxter equation and the A -constraint, preprint, arXiv: math.AG/0008156.Google Scholar
  16. Scharlau, W. (2001) On the classification of vector bundles and symmetric bilinear forms over projective varieties, preprint, Universität Münster.Google Scholar
  17. Seidel, P., and Thomas, R. P. (2000) Braid group actions on derived categories of coherent sheaves, preprint, arXiv: math.AG/0001043.Google Scholar
  18. Yudin, I. (2001) Diploma thesis, Kaiserslautern.Google Scholar

Copyright information

© Springer Science+Business Media Dordrecht 2001

Authors and Affiliations

  • I. Burban
    • 1
  • Yu. Drozd
    • 2
  • G.-M Greuel
    • 1
  1. 1.University of KaiserslauternGermany
  2. 2.Kiev Taras Shevchenko UniversityUkraine

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