Polynomials with general C^2-fibers are variables. I

Details Polynomials with general C^2-fibers are variables. I Polynomials with general C^2-fibers are variables. I

1999-12-07

arxiv.org/abs/math/9912058v1

Mathematics

Algebraic Geometry

Scientific paper

Suppose that X' is a smooth affine algebraic variety of dimension 3 with H_3(X')=0 which is a UFD and whose invertible functions are constants. Suppose that Z is 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. As a consequence we obtain the fact formulated in the title of the paper.

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