Skip to main content
SpringerLink
Log in

Faltings’ Construction of the Moduli Space of Vector Bundles on a Smooth Projective Curve

  • Conference paper
Book cover Affine Flag Manifolds and Principal Bundles

Part of the book series: Trends in Mathematics ((TM))

  • 2124 Accesses

Abstract

In 1993 Faltings gave a construction of the moduli space of semistable vector bundles on a smooth projective curve X over an algebraically closed field k. This construction was presented by the author at the German-Spanish Workshop on Moduli Spaces of Vector Bundles at the University of Essen in February 2007.

To ease notation and to simplify the necessary proofs only the case of rank two vector bundles with determinant isomorphic to ωX is considered.

These notes give a self-contained introduction to the moduli spaces of vector bundles and the generalized θ-divisor.

This is a preview of subscription content, log in via an institution to check access.

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 39.99
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 54.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 54.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. L. Álvarez-Cónsul, A. King, A functorial construction of moduli of sheaves, Invent. math. 168 (2007), 613–666.

    Article  MATH  MathSciNet  Google Scholar 

  2. W. Barth, Moduli of vector bundles on the projective plane, Invent. math. 42 (1977), 63–91.

    Article  MATH  MathSciNet  Google Scholar 

  3. A. Beauville, Vector bundles on curves and generalized theta functions: recent results and open problems, in H. Clemens (ed.) et al., Current topics in complex algebraic geometry, Cambridge Univ. Press, New York, 1995.

    Google Scholar 

  4. A. Beauville, Vector bundles on curves and theta functions, in Moduli spaces and arithmetic geometry, Adv. Stud. Pure Math. 45, Math. Soc. Japan, Tokyo, 2006, 145–156.

    Google Scholar 

  5. H. Clemens, J. Kollár, S. Mori, Higher dimensional complex geometry, Astérisque 166 (1988).

    Google Scholar 

  6. J.-M. Drezet, M.S. Narasimhan, Groupe de Picard des variétés de modules de fibrés semi-stables sur les courbes algébriques, Invent. math. 97 (1989), 53–94.

    Article  MATH  MathSciNet  Google Scholar 

  7. D. Eisenbud, Commutative Algebra, with a view toward Algebraic Geometry, Springer-Verlag, New York, 1995.

    MATH  Google Scholar 

  8. G. Faltings, Stable G-bundles and projective connections, J. Alg. Geom. 2 (1993), 507–568.

    MATH  MathSciNet  Google Scholar 

  9. A. Grothendieck, Techniques de construction et théorèmes d’existence en géométrie algébrique, IV. Les schémas des Hilbert, Séminaire Bourbaki 221 (1960/61).

    Google Scholar 

  10. R. Hartshorne, Algebraic geometry, GTM 52, New York, 1977.

    Google Scholar 

  11. G. Hein, Duality construction of moduli spaces, Geometriae Dedicata 75 (1999), 101–113.

    Article  MATH  MathSciNet  Google Scholar 

  12. G. Hein, Raynaud’s vector bundles and base points of the generalized Theta divisor, Math. Zeitschrift 257 (2007), 597–611.

    Article  MATH  MathSciNet  Google Scholar 

  13. G. Hein, Raynaud vector bundles, Monatsh. Math. 157 (2009), 233–245.

    Article  MATH  MathSciNet  Google Scholar 

  14. G. Hein, D. Ploog, Postnikov-Stability for Complexes, math.AG/0704.2512.

    Google Scholar 

  15. G. Hein, D. Ploog, Postnikov-Stability versus Semistability of Sheaves, math.AG/0901.1554.

    Google Scholar 

  16. K. Hulek, Stable rank-2 vector bundles on ℙ 2 with c 1 odd, Math. Ann. 242 (1979), 241–266.

    Article  MathSciNet  Google Scholar 

  17. D. Huybrechts, M. Lehn, The geometry of moduli spaces of shaves, Aspects of Math. E 31, Vieweg, au]Braunschweig, 1997.

    Google Scholar 

  18. S. Langton, Valuative criteria for families of vector bundles on algebraic varieties, Ann. Math. 101 (1975), 88–110.

    Article  MathSciNet  Google Scholar 

  19. J. Le Potier, Fibré déterminant et courbes de saut sur les surfaces algébriques, London Math. Soc. Lect. Note Ser. 179 (1992), 213–240.

    Google Scholar 

  20. J. Le Potier, Module des fibrés semistables et fonctions thêta, Lect. Notes in Pure and Appl. Math. 179 (1996), 83–101.

    Google Scholar 

  21. J. Li, Algebraic geometric interpretation of Donaldson’s polynomial invariants, J. Diff. Geom. 37 (1993), 417–466.

    MATH  Google Scholar 

  22. D. Mumford, Geometric invariant theory, Springer-Verlag, Berlin, 1965.

    MATH  Google Scholar 

  23. D. Mumford, Abelian Varieties, Oxford Univ. Press, London, 1970.

    MATH  Google Scholar 

  24. C. Okonek, M. Schneider, H. Spindler, Vector bundles on complex projective space, Birkhäuser, Boston, 1980.

    Google Scholar 

  25. 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.

    Article  MATH  MathSciNet  Google Scholar 

  26. M. Raynaud, Section des fibrés vectoriels sur une courbe, Bull. Soc. Math. France 110 (1982), 103–125.

    MATH  MathSciNet  Google Scholar 

  27. S.S. Shatz, The decomposition and specialisation of algebraic families of vector bundles, Comp. Math. 35 (1977), 163–187.

    MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Universität Duisburg-Essen, Fakultät Mathematik, D-45117, Essen, Germany

    Georg Hein

Authors
  1. Georg Hein
    View author publications

    You can also search for this author in PubMed Google Scholar

Editor information

Editors and Affiliations

  1. Institut für Mathematik, Freie Universität Berlin, Arnimallee 3, 14195, Berlin, Germany

    Alexander Schmitt

Rights and permissions

Reprints and permissions

Copyright information

© 2010 Springer Basel AG

About this paper

Cite this paper

Hein, G. (2010). Faltings’ Construction of the Moduli Space of Vector Bundles on a Smooth Projective Curve. In: Schmitt, A. (eds) Affine Flag Manifolds and Principal Bundles. Trends in Mathematics. Springer, Basel. https://doi.org/10.1007/978-3-0346-0288-4_3

Download citation

Publish with us

Policies and ethics

Search

Navigation