The geometry of sporadic \(\mathbb{C}^\ast\)-embeddings into \(\mathbb{C}^2\) (Q286551)

There is a common feeling that singularity resolution technique is important in affine algebraic geometry and may provide some deep insights. Different authors (including Abhyankar, Vitushkin, Kishimoto, Scholze, Gizatullin and authors of the paper) worked in this area and got inspiring ideas. The paper follows that line.NEWLINENEWLINEA closed algebraic embedding of \(\mathbb{C}^*=\mathbb{C}^1\setminus\{0\}\) into \(\mathbb{C}^2\) is \textit{sporadic} if for every curve \(A\subseteq \mathbb{C}^2\) isomorphic to an affine line the intersection with \(\mathbb{C}^*\) is at least \(2\). Non-sporadic embeddings have been classified. The authors establish geometric and algebraic tools to classify them based on the analysis of the minimal log resolution \((X,D)\to (\mathbb{P}^2,U)\), where \(U\) is the closure of \(\mathbb{C}^*\) on \(\mathbb{P}^2\). They show in particular that one can choose coordinates on \(\mathbb{C}^2\) in which the type at infinity of the \(\mathbb{C}^*\) and the self-intersection of its proper transform on \(X\) are sharply limited.NEWLINENEWLINEThe main technique related with singularity resolutions in fact.