Very general monomial valuations of P2 and a Nagata type conjecture

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.