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Israel Journal of Mathematics

, Volume 116, Issue 1, pp 323–343 | Cite as

Uniform Zariski’s theorem on fundamental groups

  • Shulim Kaliman
Article
  • 40 Downloads

Abstract

The Zariski theorem says that for every hypersurface in a complex projective (resp. affine) space and for every generic plane in the projective (resp. affine) space the natural embedding generates an isomorphism of the fundamental groups of the complements to the hypersurface in the plane and in the space. If a family of hypersurfaces depends algebraically on parameters then it is not true in general that there exists a plane such that the natural embedding generates an isomorphism of the fundamental groups of the complements to each hypersurface from this family in the plane and in the space. But we show that in the affine case such a plane exists after a polynomial coordinate substitution.

Keywords

Fundamental Group Algebraic Variety Intersection Number Simple Loop Natural Embedding 
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.

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References

  1. [1]
    Sh. Kaliman,On the Jacobian conjecture, Proceedings of the American Mathematical Society117 (1993), 45–51.MATHCrossRefMathSciNetGoogle Scholar
  2. [2]
    E. van Kampen,On the connection between the fundamental groups of some related spaces, American Journal of Mathematics55 (1933), 261–267.MATHGoogle Scholar
  3. [3]
    O. Zariski,A theorem on the Poincaré group of an algebraic hypersurface, Annals of Mathematics38 (1937), 131–141.CrossRefMathSciNetGoogle Scholar

Copyright information

© Hebrew University 2000

Authors and Affiliations

  1. 1.Department of Mathematics and Computer ScienceUniversity of MiamiCoral GablesUSA

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