Counting Singularities of Quadratic Forms on Vector Bundles

  • Wolf Barth
Part of the Progress in Mathematics book series (PM, volume 7)


The study of surfaces in ℙ3, with many nodes (= ordinary double points) is a beautiful classical topic, which recently found much attention again [3, 4]. All systematic ways to produce such surfaces seem related to symmetric matrices of homogeneous polynomials or, more generally, to quadratic forms on vector bundles: If the form q on the bundle E is generic, then q is of maximal rank on an open set. The rank of q is one less on the discriminant hypersurface det q = o, which represents the class 2c1. (E*). This hypersurface is nonsingular in codimension one, but has ordinary double points in codimension two exactly where rank q drops one more step.


Quadratic Form Exact Sequence Vector Bundle Chern Class Divisor Class 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [1]
    Barth, W. Some properties of stable rank-2 vector bundles on ℙn. Math. Ann. 226, 125–150 (1977).MathSciNetMATHCrossRefGoogle Scholar
  2. [2]
    Barth, W. Kummer surfaces associated with the Mumford Horrocks bundle, talk at Angers (1979).Google Scholar
  3. [3]
    Beauville, A. Surfaces with many nodes (¼(5) = 31), talk at Angers (1979).Google Scholar
  4. [4]
    Catanese, F. Babbage’s conjecture, contact of surfaces, symmetrical determinant varieties and applications, preprint, Pisa, (1979).Google Scholar
  5. [5]
    Gallarati, D. Intorno ad una superficie del sesto ordine avente 63 nodi. Boll. U.M.I. Serie III, Anno VII, 392–396 (1952).MathSciNetGoogle Scholar
  6. [6]
    Gallarati, D. Ri cherche sul contatto di superficie algebriche lungo curve. Mem. Acad. royal Belg. XXXII fasc. 3 (1960).Google Scholar
  7. [7]
    Hartshorne, R. Algebraic Geometry, Springer Verlag (1977).Google Scholar
  8. [8]
    Hartshorne, R. Stable vector bundles of rank 2 on ℙ3. Math. Ann. 238, 229–280 (1978).MathSciNetMATHCrossRefGoogle Scholar
  9. [9]
    Porteous, I.R. Simple singularities of maps. Springer Lecture Notes in Mathematics 192, 286–307 (1971).MathSciNetCrossRefGoogle Scholar
  10. [10]
    Togliatti, E. Una notevole superficie di; 5° ordine con soli punti doppi isolati, Festschrift R. Fueter, Zurich (1940), 127–135.Google Scholar
  11. [11]
    Vogelaar, J.A. Constructing vector bundles from codimension-two subvarieties. proefschrift, Leiden (1978).Google Scholar
  12. [12]
    Catanese, F. Ceresa, G. Constructing sextic surfaces with a given number of nodes, preprint (1979).Google Scholar

Note added in proof

  1. Loring Tu: Variation of Hodge Structure and the Local Torelli Problem. Thesis Harvard 1979.Google Scholar
  2. T. Jozefiak, A. Lascoux, P. Pragacz: Classes of determinantal varieties associated with symmetric and antisymmetric matrices.Google Scholar

Copyright information

© Birkhäuser, Boston 1980

Authors and Affiliations

  • Wolf Barth
    • 1
  1. 1.Mathematisches InstitutUniversität Erlangen-NürnbergErlangenWest Germany

Personalised recommendations