An effective criterion for algebraicity of rational normal surfaces

We give a novel and effective criterion for algebraicity of rational normal analytic surfaces constructed from resolving the singularity of an irreducible curve-germ on $CP^2$ and contracting the strict transform of a given line and all but the `last' of the exceptional divisors. As a by-product we construct a new class of analytic non-algebraic rational normal surfaces which are `very close' to being algebraic. These results are local reformulations of some results in (Mondal, 2011) which sets up 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. This article is meant partly to be an exposition to (Mondal, 2011) and we give a proof of the correspondence theorem of (Mondal, 2011) in the `first non-trivial case'.