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

Abstract

A closed algebraic embedding of ${\mathbb{C}}^{\ast }={\mathbb{C}}^1\setminus \{0\}$ into ${\mathbb{C}}^2$ is 'sporadic' if for every curve $A\subseteq {\mathbb{C}}^2$ isomorphic to an affine line the intersection with ${\mathbb{C}}^{\ast }$ 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 ${\mathbb{C}}^{\ast }$ 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 ${\mathbb{C}}^{\ast }$ and the self-intersection of its proper transform on $X$ are sharply limited.

34 pages, 1 figure

References in corpus (1)

• Complex algebraic curves. Annuli

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