Primitive normal completions of the affine plane II

TL;DR

This paper investigates primitive normal analytic compactifications of the complex plane, establishing criteria for their projectivity and algebraicity based on associated jets of curve germs at infinity, and demonstrating the existence of non-algebraic examples.

Contribution

It provides explicit criteria to determine when primitive compactifications are projective or algebraic, extending previous work on normal analytic compactifications of a5^2.

Findings

Primitive compactifications can be non-algebraic.

Projectivity is equivalent to algebraicity for these compactifications.

Criteria based on jets of curve germs determine their algebraic nature.

Abstract

In this article we continue from \cite{sub2-1} the study of normal analytic compactifications of $\cc^2$ from the point of view of their associated pencils of jets of curve germs centered at infinity. If $\bar X$ is a normal analytic compactification of $\cc^2$ which is {\em primitive}, i.e.\ $\bar X \setminus \cc^2$ is irreducible curve, then we show that $\bar X$ is projective iff $\bar X$ is algebraic iff at least one of the jets in the associated pencil of jets of curve-germs can be represented by a planar curve with one place at infinity. As a result we show that there are primitive normal analytic compactifications of $\cc^2$ which are {\em not} algebraic. We give explicit criteria for determining if the primitive compactification corresponding to a jet of curve germs at infinity is projective or not.