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The structure of algebraic embeddings of C² into C³ (The normal quartic hypersurface case. I)

We obtain the affirmative answer for a special case of the line ar zation problem for algebraic embeddings of C2 into C3. Indeed, we determine all the compactifications(X, Y) of C2 such thatX are normal quartic hypersurfaces in P3 without triple points andY are hyperplane sections of X. Moreover, for each(X, Y), we construct a tame automorphism of C3 which transforms the hypersurface X n Y onto a coordinate hyperplane.