Algebraicity of normal analytic compactifications of $\mathbb{C}^2$ with one irreducible curve 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 byproduct 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 [1936] to study valuations in the “local” setting — which also extends the notion of approximate roots of polynomials considered by Abhyankar and Moh [19 73].