Transcendental versions in C n of the Nagata conjecture

Abstract

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 $\ge$ 10 general points in the projective plane P 2 with multiplicities at least l at every point, satisfies the inequality d > $\sqrt$ r $\times$ 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 $\ge$ 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.