### 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.

Original language | English (US) |
---|---|

Pages (from-to) | 323-343 |

Number of pages | 21 |

Journal | Israel Journal of Mathematics |

Volume | 116 |

State | Published - 2000 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

*Israel Journal of Mathematics*,

*116*, 323-343.

**Uniform Zariski's theorem on fundamental groups.** / Kaliman, Shulim.

Research output: Contribution to journal › Article

*Israel Journal of Mathematics*, vol. 116, pp. 323-343.

}

TY - JOUR

T1 - Uniform Zariski's theorem on fundamental groups

AU - Kaliman, Shulim

PY - 2000

Y1 - 2000

N2 - 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.

AB - 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.

UR - http://www.scopus.com/inward/record.url?scp=0034355884&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0034355884&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0034355884

VL - 116

SP - 323

EP - 343

JO - Israel Journal of Mathematics

JF - Israel Journal of Mathematics

SN - 0021-2172

ER -