The structure of algebraic embeddings of \(\mathbb{C}^2\) into \(\mathbb{C}^3\) (the normal quartic hypersurface case. I) (Q5949465)

Since the works of S. S. Abhyankar, T. T. Moh and of M. Suzuki in the 1970's, who treated the case \(n= 1\), it is conjectured that every algebraic embedding \(f: \mathbb{C}^n\to \mathbb{C}^{n+1}\) is equivalent to a linear embedding, namely there exists an algebraic automorphism \(\Phi\) of \(\mathbb{C}^{n+1}\) such that \(\Phi\circ f\) is a linear embedding. Even for \(n= 2\), the conjecture is wide open in full generality. In a previous work [Kyushu J. Math. 53, No. 1, 67-106 (1999; Zbl 0937.14008)], making use of some classification material and of so-called Nagata automorphisms, the author gave explicit straightenings \(\Phi\) for every algebraic embedding \(f\) of degree less than or equal to three.NEWLINENEWLINENEWLINELet \(X_f\) be the closure of \(f(\mathbb{C}^2)\) in \(P_3(\mathbb{C})\). In the paper under review, the author endeavours the study of the conjecture for such degree four projective surfaces \(X_f\), which he divides in three subcases. (I) \(X_f\) is normal and it has at least a triple point; (II) \(X_f\) is normal and it has no triple point; (III) \(X_f\) is non-normal.NEWLINENEWLINENEWLINEThe author achieves to provide fourteen explicit normal forms for such \(X_f\) in subcase (I) and he also provides explicit straightenings \(\Phi\). Subcases (II) and (III) should be dealt with elsewhere.