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Embeddings of a family of Danielewski hypersurfaces and certain C + -actions on C 3

Lucy Moser-JauslinPierre-Marie Poloni — 2006

Annales de l’institut Fourier

We consider the family of polynomials in C [ x , y , z ] of the form x 2 y - z 2 - x q ( x , z ) . Two such polynomials P 1 and P 2 are equivalent if there is an automorphism ϕ * of C [ x , y , z ] such that ϕ * ( P 1 ) = P 2 . We give a complete classification of the equivalence classes of these polynomials in the algebraic and analytic category. As a consequence, we find the following results. There are explicit examples of inequivalent polynomials P 1 and P 2 such that the zero set of P 1 + c is isomorphic to the zero set of P 2 + c for all c C . There exist polynomials which are algebraically...

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