The structure of algebraic embeddings of $\mathbb{C}^{2}$ into $\mathbb{C}^{3}$ (the normal quartic hypersurface case. II)

Abstract

We obtain the affirmative answer for a special case of the linearization problem for algebraic embeddings of $\mathbb{C}^{2}$ into $\mathbb{C}^{3}$. Indeed, we determine all the compactifications $(X,Y)$ of $\mathbb{C}^{2}$ such that $X$ are normal quartic hypersurfaces in $\mathbb{P}^{3}$ without triple points and $Y$ are hyperplane sections of $X$. Moreover, for each $(X,Y)$, we construct a tame automorphism of $\mathbb{C}^{3}$ which transforms the hypersurface $X\setminus Y$ onto a coordinate hyperplane.