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).
Keywords
Cite
@article{arxiv.1110.6905,
title = {Analytic Compactifications of C^2 part I - curvettes at infinity},
author = {Pinaki Mondal},
journal= {arXiv preprint arXiv:1110.6905},
year = {2015}
}
Comments
Final and substantially improved version, accepted in the Mathematical Reports of the Academy of Science, Royal Society of Canada