no. 1
Transcendental versions in ℂ n of the Nagata conjecture
[Versions transcendentales dans ℂ n de la conjecture de Nagata]
Nivoche, Stéphanie1
1 Université Côte d’Azur Laboratoire J.A. Dieudonné, CNRS UMR 7351 Université de Nice Sophia Antipolis Parc Valrose 06108 Nice cedex 2 (France)
Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 27-52.

La conjecture de Nagata est l’un des problèmes ouverts les plus intriguants dans le domaine des courbes du plan complexe. Elle s’énonce simplement. En effet, elle affirme que le plus petit degré d d’une courbe plane passant par r≥10 points généraux dans le plan projectif ℙ 2 avec des multiplicités au moins l en chaque point, satisfait l’inégalité d>r·l. Cette conjecture a été vérifiée par M. Nagata en 1959, si r est un carré parfait strictement supérieur à 9. Jusqu’à présent, elle est restée ouverte pour tout entier r≥10 non carré, après plus d’un demi-siècle d’attention de la part de nombreux chercheurs.

Dans cet article, nous formulons de nouvelles versions transcendentales de cette conjecture issues de la théorie du pluripotentiel, et qui sont équivalentes à une version dans ℂ n de la conjecture Nagata.

The Nagata Conjecture is one of the most intriguing open problems in the area of curves in the plane. It is easily stated. Namely, it predicts that the smallest degree d of a plane curve passing through r≥10 general points in the projective plane ℙ 2 with multiplicities at least l at every point, satisfies the inequality d>r·l. This conjecture has been proven by M. Nagata in 1959, if r is a perfect square greater than 9. Up to now, it remains open for every non-square r≥10, after more than a half century of attention by many researchers.

In this paper, we formulate new transcendental versions of this conjecture coming from pluripotential theory and which are equivalent to a version in ℂ n of the Nagata Conjecture.

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DOI : 10.5802/aif.3402
Classification : 14H50, 32U10, 32U25, 32U35, 32U40, 32W20
Keywords: Nagata conjecture, pluripotential theory, pluricomplex Green function
Mot clés : Conjecture de Nagata, théorie du pluripotentiel, fonction de Green pluricomplexe

Nivoche, Stéphanie 1

1 Université Côte d’Azur Laboratoire J.A. Dieudonné, CNRS UMR 7351 Université de Nice Sophia Antipolis Parc Valrose 06108 Nice cedex 2 (France)
Licence : CC-BY-ND 4.0
Droits d'auteur : Les auteurs conservent leurs droits 
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Nivoche, Stéphanie. Transcendental versions in ${{\protect \mathbb{C}}}^n$ of the Nagata conjecture. Annales de l'Institut Fourier, Tome 71 (2021) no. 1, pp. 27-52. doi : 10.5802/aif.3402. https://aif.centre-mersenne.org/articles/10.5802/aif.3402/

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