Algebraicity of normal analytic compactifications of C^2 with one irreducible curve at infinity

TL;DR

This paper provides an effective criterion to determine when a normal analytic compactification of C^2 with one irreducible curve at infinity is algebraic, establishing a correspondence with algebraic curves and constructing examples with different algebraic properties.

Contribution

It introduces a criterion based on key forms to distinguish algebraic from non-algebraic compactifications and links these to algebraic curves with one place at infinity.

Findings

Established a criterion for algebraicity of compactifications.

Constructed pairs of surfaces with different algebraic properties.

Linked compactifications to algebraic curves with one place at infinity.

Abstract

We present an effective criterion to determine if a normal analytic compactification of C^2 with one irreducible curve at infinity is algebraic or not. As a by product we establish a correspondence between normal algebraic compactifications of C^2 with one irreducible curve at infinity and algebraic curves contained in C^2 with one place at infinity. Using our criterion we construct pairs of homeomorphic normal analytic surfaces with minimally elliptic singularities such that one of the surfaces is algebraic and the other is not. Our main technical tool is the sequence of "key forms" - a 'global' variant of the sequence of "key polynomials" introduced by MacLane to study valuations in the 'local' setting - which also extends the notion of "approximate roots" of polynomials considered by Abhyankar and Moh.