An effective criterion for algebraic contractibility of rational curves

TL;DR

This paper provides a precise criterion to determine when certain rational surfaces obtained by contracting specific divisors are algebraic, involving global key polynomials, and constructs examples of non-algebraic Moishezon surfaces.

Contribution

It introduces an effective necessary and sufficient criterion for algebraicity of contracted surfaces using global key polynomials, and constructs non-algebraic Moishezon surfaces with simple singularities.

Findings

Established a criterion linking algebraicity to key polynomials.

Constructed non-algebraic Moishezon surfaces with minimal singularities.

Connected algebraic compactifications of C^2 with curves at infinity.

Abstract

Let f: Y -> CP^2 be a birational morphism of non-singular (rational) surfaces. We give an effective (necessary and sufficient) criterion for algebraicity of the surfaces resulting from contraction of the union of the strict transform of a line on CP^2 and all but one of the exceptional divisors of f. As a by-product we construct normal non-algebraic Moishezon surfaces with the `simplest possible' singularities, which in particular completes the answer to a remark of Grauert. Our criterion involves `global variants' of `key polynomials' introduced by MacLane. The geometric formulation of the criterion yields a correspondence between normal algebraic compactifications of C^2 with one irreducible curve at infinity and algebraic curves in C^2 with one place at infinity.