Very general monomial valuations of P2 and a Nagata type conjecture

Agraïments: This research was supported through the programme "Research in Pairs" by the Mathematisches Forschungsinstitut Oberwolfach in 2013. Alex Küronya was partially supported by the DFG-Forschergruppe 790 "Classification of Algebraic Surfaces and Compact Complex Manifolds", by the DFG-Graduiertenkolleg 1821 "Cohomological Methods in Geometry", and by the OTKA grants 77476 and 81203 of the Hungarian Academy of Sciences. Brian Harbourne's work on this project was partially sponsored by the National Security Agency under Grant/Cooperative agreement "Advances on Fat Points and Symbolic Powers," Number H98230-11-1-0139. The United States Government is authorized to reproduce and distribute reprints notwithstanding any copyright notice. 245 10It 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