Annali di Matematica Pura ed Applicata

, Volume 56, Issue 1, pp 359–373 | Cite as

The tangent direction bundle of an algebraic variety and generalized Jacobians of linear systems

  • A. W. Ingleton
  • D. B. Scott


It is well-known that, on an algebraic variety V of dimension d, there is associated with a set of linear systems whose total dimension is d a Jacobian variety (of dimension d−1) at any point of which (other than base points of the linear systems) there is at least one line (formally) tangent to every variety of each system which passes through the point. This notion generalizes to a set of linear systems of total dimension d+r (0≤r<d), the generalized Jacobian being then of dimension d−r−1. The final aim of this paper is to obtain a general formula (Theorem 5.2) for the homology class of this generalized Jacobian. The proof is derived with the aid of cohomological and bundle-theoretic methods from the study of the tangent direction bundle of V, and the earlier part of the paper establishes the necessary techniques (which are not without their independent interest) for our purposes.


Linear System General Formula Base Point Variety Versus Early Part 
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.


  1. [1]
    M. Baldassarri,Algebraic Varieties « Ergebnisse der Math. », Berlin, 1956.Google Scholar
  2. [2]
    A. Borel,Sur la cohomologie des espaces fibrés principaux et des espaces homogènes de groupes de Lie compacts, « Ann. Math. » (2), 57 (1953), 115–207.zbMATHMathSciNetCrossRefGoogle Scholar
  3. [3]
    A. Borel andF. Hirzebruch,Characteristic classes and homogeneous spaces I, « Amer. J. Math. » 80 (1958), 458–538;MathSciNetGoogle Scholar
  4. [3a]
    —— ——,Characteristic classes and homogeneous spaces II, ibid. « 81 (1959), 315–382.MathSciNetGoogle Scholar
  5. [4]
    S. S. Chern,On the characteristic classes of complex sphere bundles and algebraic varieties, « Amer. J. Math. » 75 (1953), 565–597.zbMATHMathSciNetGoogle Scholar
  6. [5]
    C. Ehresmann,Sur la topologie de certains espaces homogènes, « Ann. Math. » (2), 35 (1934), 396–443.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [6]
    F. Hirzebruch,Neue Topologische Methoden in der Algebraischen Geometrie « Ergebnisse der Math. », Berlin, 1956.Google Scholar
  8. [7]
    A. W. Ingleton,The tangent flag bundle of an algebraic variety in preparation.Google Scholar
  9. [8]
    D. Monk,The geometry of flag manifolds, « Proc. London Math. Soc. » (3), 9 (1959), 253–286.zbMATHMathSciNetGoogle Scholar
  10. [9]
    D. B. Scott,Tangent-direction bundles of algebraic varieties, « Proc. London Math. Soc. » (3), 11 (1961), 57–79.zbMATHMathSciNetGoogle Scholar
  11. [10]
    F. Severi,Fondamenti per la geometria sulle varietà algebriche, « Ann. Mat. Pura Appl. » (4), 32 (1951), 1–81.zbMATHMathSciNetGoogle Scholar
  12. [11]
    N. Steenrod,The Topology of Fibre Bundles, Princeton, 1951.Google Scholar

Copyright information

© Nicola Zanichelli Editore 1961

Authors and Affiliations

  • A. W. Ingleton
    • 1
  • D. B. Scott
    • 1
  1. 1.LondonEngland

Personalised recommendations