32C20 — 1 articles found.

Analytic Compactifications of $C^2$ Part I—Curvettes at Infinity

C. R. Math. Rep. Acad. Sci. Canada Vol. 38 (2) 2016, pp. 41-74
Vol.38 (2) 2016
Pinaki Mondal Details
(Received: 2015-02-10 , Revised: 2015-07-09 )
(Received: 2015-02-10 , Revised: 2015-07-09 )

Pinaki Mondal,The School of Mathematics, Physics & Technology, College of The Bahamas, Nassau, Bahamas; e-mail: pinakio@gmail.com

Abstract/Résumé:

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).

Keywords: Compactifications of $C^2$, curvettes, discreet valuations., polynomial automorphisms

AMS Subject Classification: Rational and ruled surfaces, , Normal analytic spaces 14J26, 14M27, 32C20

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