Very general monomial valuations of ${\mathbb{P}}^2$ and a Nagata type conjecture

Abstract

It is well known that multi-point Seshadri constants for a small number $s$ of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for $s\ge 9$ points. Tackling the problem in the language of valuations one can make sense of $s$ points for any positive real $s\ge 1$. We show somewhat surprisingly that a Nagata-type conjecture should be valid for $s\ge 8+1/36$ points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for $s\le 7+1/9$.

25 pages, 2 figures. Updated version of the Oberwolfach Preprint OWP 2013-22, with some new material. Comments welcome