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 t of points in the projective plane are submaximal. It is predicted by the Nagata conjecture that their values are maximal for t≥9 points. Tackling the problem in the language of valuations one can make sense of t points for any real t≥1. We show somewhat surprisingly that a Nagata-type conjecture should be valid for t≥8+1/36 points and we compute explicitly all Seshadri constants (expressed here as the asymptotic maximal vanishing element) for t≤7+1/9. In the range 7+1/9≤t≤8+1/36 we are able to compute some sporadic values.