### Abstract

Let X′ be a complex affine algebraic threefold with H_{3}(X′) = 0 which is a UFD and whose invertible functions are constants. Let Z be a Zariski open subset of X′ which has a morphism p : Z → U into a curve U such that all fibers of p are isomorphic to C^{2}. We prove that X′ is isomorphic to C^{3} iff none of irreducible components of X′ \ Z has non-isolated singularities. Furthermore, if X′ is C^{3} then p extends to a polynomial on C^{3} which is linear in a suitable coordinate system. This implies the fact formulated in the title of the paper.

Original language | English (US) |
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Pages (from-to) | 161-190 |

Number of pages | 30 |

Journal | Pacific Journal of Mathematics |

Volume | 203 |

Issue number | 1 |

State | Published - 2002 |

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### ASJC Scopus subject areas

- Mathematics(all)

### Cite this

^{2}-fibers are variables.

*Pacific Journal of Mathematics*,

*203*(1), 161-190.

**Polynomials with general C ^{2}-fibers are variables.** / Kaliman, Shulim.

Research output: Contribution to journal › Article

^{2}-fibers are variables',

*Pacific Journal of Mathematics*, vol. 203, no. 1, pp. 161-190.

^{2}-fibers are variables. Pacific Journal of Mathematics. 2002;203(1):161-190.

}

TY - JOUR

T1 - Polynomials with general C2-fibers are variables

AU - Kaliman, Shulim

PY - 2002

Y1 - 2002

N2 - Let X′ be a complex affine algebraic threefold with H3(X′) = 0 which is a UFD and whose invertible functions are constants. Let Z be a Zariski open subset of X′ which has a morphism p : Z → U into a curve U such that all fibers of p are isomorphic to C2. We prove that X′ is isomorphic to C3 iff none of irreducible components of X′ \ Z has non-isolated singularities. Furthermore, if X′ is C3 then p extends to a polynomial on C3 which is linear in a suitable coordinate system. This implies the fact formulated in the title of the paper.

AB - Let X′ be a complex affine algebraic threefold with H3(X′) = 0 which is a UFD and whose invertible functions are constants. Let Z be a Zariski open subset of X′ which has a morphism p : Z → U into a curve U such that all fibers of p are isomorphic to C2. We prove that X′ is isomorphic to C3 iff none of irreducible components of X′ \ Z has non-isolated singularities. Furthermore, if X′ is C3 then p extends to a polynomial on C3 which is linear in a suitable coordinate system. This implies the fact formulated in the title of the paper.

UR - http://www.scopus.com/inward/record.url?scp=0036347427&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=0036347427&partnerID=8YFLogxK

M3 - Article

AN - SCOPUS:0036347427

VL - 203

SP - 161

EP - 190

JO - Pacific Journal of Mathematics

JF - Pacific Journal of Mathematics

SN - 0030-8730

IS - 1

ER -