The geometry of sporadic $\mathbb{C}^*$-embeddings into $\mathbb{C}^2$

TL;DR

This paper investigates the rare class of sporadic algebraic embeddings of the punctured complex line into the plane, developing geometric and algebraic tools to classify them based on their minimal log resolutions.

Contribution

It introduces new methods for classifying sporadic embeddings using minimal log resolutions and analyzes their geometric properties at infinity.

Findings

Sporadic embeddings are sharply limited by their type at infinity.

The self-intersection of the proper transform constrains possible embeddings.

Tools are established for future classification of these rare embeddings.

Abstract

A closed algebraic embedding of $\mathbb{C}^*=\mathbb{C}^1\setminus\{0\}$ into $\mathbb{C}^2$ is '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. There are very few known sporadic embeddings. We 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 $\mathbb{P}^2$. We 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.