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Analytic Compactifications of $C^2$ Part I—Curvettes at Infinity
C. R. Math. Rep. Acad. Sci. Canada Vol. 38 (2) 2016, pp. 41-74
Pinaki Mondal (Received: 2015/02/10, Revised: 2015/07/09)

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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 (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 \(P^2\) 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 (2004).

Nous étudions les compactifications analytiques normales de \(C^2\) et décrivons leurs singularités et la configuration des courbes à l’infini, en particulier ameliorant et généralisant les résultats de Brenton (1973). Comme un sous-produit, nous donnons de nouvelles preuves du théorème de Jung sur les automorphismes polynomiaux de \(C^2 \) et le résultat de Remmert et Van de Ven que \(P^2\) est la seule compactification analytique lisse de \(C^2\) pour laquelle la courbe à l’infini est irréductible. Nous donnons aussi une réponse complète à la question de l’existence de compactifications de \(C^2 \) avec des valorisations divisorielles préscrites à l’infini. En particulier, nous montrons qu’une évaluation sur \(C(x,y) \) centrée à l’infini détermine une compactification de \(C^2\) ssi elle est positivement asymétrique dans le sens de Favre and Jonsson (2004).

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