On algebraic and non-algebraic neighborhoods of rational curves

TL;DR

This paper constructs examples of neighborhoods of rational curves in complex surfaces that are non-algebraic yet have rich meromorphic function fields, revealing nuanced distinctions between algebraic and analytic embeddings.

Contribution

It provides explicit constructions of non-algebraic neighborhoods of rational curves with specified self-intersection, and classifies algebraic germs of such embeddings.

Findings

Existence of non-algebraic neighborhoods with transcendence degree 2

Two methods of construction: blowdowns and ramified coverings

Classification of algebraic germs of embeddings of P^1

Abstract

We prove that for any $d>0$ there exists an embedding of the Riemann sphere $\mathbb P^1$ in a smooth complex surface, with self-intersection $d$, such that the germ of this embedding cannot be extended to an embedding in an algebraic surface but the field of germs of meromorphic functions along $C$ has transcendence degree $2$ over $\mathbb C$. We give two different constructions of such neighborhoods, either as blowdowns of a neighborhood of the smooth plane conic, or as ramified coverings of a neighborhood of a hyperplane section of a surface of minimal degree. The proofs of non-algebraicity of these neighborhoods are based on a classification, up to isomorphism, of algebraic germs of embeddings of $\mathbb P^1$, which is also obtained in the paper.