Analytic Compactifications of C^2 part I - curvettes at infinity

Abstract

We study normal analytic compactifications of C^2 and describe their singularities and configuration of curves at infinity, in particular improving and generalizing results of (Brenton, Math. Ann. 206:303--310, 1973). As a by product we give new proofs of Jung's theorem on polynomial automorphisms of C^2 and Remmert and Van de Ven's result that CP^2 is the only smooth analytic compactification of C^2 for which the curve at infinity is irreducible. We also give a complete answer to the question of existence of compactifications of C^2 with prescribed divisorial valuations at infinity. In particular, we show that a valuation on C(x,y) centered at infinity determines a compactification of C^2 iff it is "positively skewed" in the sense of (Favre and Jonsson, Ann. Sci. Ecole Norm. Sup. 40(2):309--349, 2007).

Final and substantially improved version, accepted in the Mathematical Reports of the Academy of Science, Royal Society of Canada

Cited by in corpus (4)

• How to determine the sign of a valuation on C[x,y]?
• Intersection of valuation rings in $k[x,y]$
• An effective criterion for algebraic contractibility of rational curves
• Algebraicity of normal analytic compactifications of C^2 with one irreducible curve at infinity