A remark on Abhyankar's space lines
An (n,n-1,n-2)-line \(C_ n\) in \({\mathbb{A}}^ 3\) is the image of \({\mathbb{A}}^ 1\) under the map \((f_ 0(t),f_ 1(t),f_ 2(t))\) where \(\deg (f_ i)=n-i\). \(C_ n\) is said to be rectifiable if there exists an automorphism \(\Phi_ n\) of \({\mathbb{A}}^ 3\) such that \(\Phi_ n(C_ n)\) is a straight line. The author proves the rectifiability of the curve \(C_ 6: (t+t^ 6,t^ 5,t^ 4)\). The automorphism \(\Phi\) he obtains should not be tame (i.e. the product of linear and triangular automorphisms), according to a conjecture of \textit{M. Nagata} [cf. ``On automorphism group of k[X,Y]'' (Tokyo 1972; Zbl 0306.14001)]. On the other hand, the author gives an example of another (6,5,4)-line \(C'_ 6\) which is tamely rectifiable, showing that the possible existence of a tame automorphism that rectifies \(C_ 6\) cannot be excluded simply from the triple (6,5,4).
scientific article; zbMATH DE number 4150388