Some properties of \(\mathbb C^*\) in \({\mathbb C^2}\) (Q2850560)

It is known by the theorem of \textit{S. S. Abhyankar} and \textit{T. T. Moh} [J. Reine Angew. Math. 276, 148--166 (1975; Zbl 0332.14004)] and \textit{M. Suzuki} [J. Math. Soc. Japan 26, 241--257 (1974; Zbl 0276.14001)], that if we embed algebraically the complex line \(\mathbb{C}^1\) into the complex plane \(\mathbb{C}^2\) then it is possible to choose coordinates \((x,y)\) on the plane so that the image is described by the equation \(x=0\). It is a natural and interesting question what is the situation for \(\mathbb{C}^*=\mathbb{C}^1-\{0\}\). It is clearly more difficult, for example the embeddings \(t\to (t^n,1/t^m)\) with \(n>m\) coprime are different from each other.NEWLINENEWLINE{Question:} Can one classify closed embeddings of \(\mathbb{C}^*\) into \(\mathbb{C}^2\) up to a choice of coordinates on \(\mathbb{C}^2\)?NEWLINENEWLINEIn [Isr. J. Math. 175, 301--347 (2010; Zbl 1202.14031)] \textit{M. Borodzik} and \textit{H. Żołądek} classified embeddings for which some regularity condition holds. Deciding whether this condition holds in general seems to be very difficult. Independently, \textit{M. Koras} and \textit{P. Russell} started the program of an unconditional classification. In their paper with \textit{P. Cassou-Nogues} [J. Algebra 322, No. 9, 2950--3002 (2009; Zbl 1218.14019)] they classified embeddings which admit a \textit{good asymptote}. If \(U\subseteq \mathbb{C}^2\) is a closed curve isomorphic to \(\mathbb{C^*}\) and \(L\subseteq \mathbb{C}^2\) is isomorphic to \(\mathbb{C}^1\) we say that \(L\) is a \textit{good asymptote} for \(U\) if \(L\cdot U\leq 1\) (the intersection counted in \(\mathbb{C}^2\)). Embeddings with a good asymptote disjoint from \(U\) (\textit{a very good asymptote}) have been already classified by \textit{S. Kaliman} in [Pac. J. Math. 174, No. 1, 141--194 (1996; Zbl 0868.32010)].NEWLINENEWLINEWhat remains is to classify closed \(\mathbb{C}^*\) embeddings into \(\mathbb{C}^2\) which do not have a good asymptote. Here the authors make a step towards this goal by showing that for such embeddings one can separate two branches at infinity of \(U\) by an automorphism of \(\mathbb{C}^2\). Equivalently, we can complete \(\mathbb{C}^2\) to \(\mathbb{P}^2\) so that the closure of \(U\) is a bi-cuspidal curve and the line at infinity meets this closure only in the two cusps. Note that by a result of \textit{M. Koras} [CRM Proceedings and Lecture Notes 54, 165--191 (2011; Zbl 1252.14042)] \(\mathbb{C}^*\) in \(\mathbb{C}^2\) can be transformed into a line by a birational map of \(\mathbb{C}^2\).NEWLINENEWLINEFor the entire collection see [Zbl 1266.14004].