The structure of algebraic embeddings of \(\mathbb{C}^{2}\) into \(\mathbb{C}^{3}\) (the normal quartic hypersurface case. II) (Q2389200)

The paper is concerned with the following conjecture: Conjecture. Let \(f:\mathbb{C}^n\rightarrow \mathbb{C}^{n+1}\) be an injective polynomial map such that \(V:=f(\mathbb{C}^n)\) is a smooth algebraic subvariety of \(\mathbb{C}^{n+1}\). Then there exists an automorphism \(\varphi\) of \(\mathbb{C}^n\) that rectifies \(V\), i.e. \(\varphi(V)=\mathbb{C}^n\times \{0\}\). The case \(n=1\) is the celebrated Abhyankar-Moh-Suzuki theorem. The paper under review proves a special case of \(n=2\). For this, the author defines \(X\) as the closure of \(f(\mathbb{C}^2)\) in \(\mathbb{P}^3\) (under the canonical embedding of \(\mathbb{C}^3\) in \(\mathbb{P}^3\)), and \(Y:=X\backslash f(\mathbb{C}^2)\). In the case that \(X\) is a normal quartic hypersurface without triple points, he proves the conjecture and describe how to find a rectifying automorphism. [For part I, cf. Osaka J. Math. 38, No. 3, 507--532 (2001; Zbl 1010.32012)].