Abstract. Inspired by results of Guardo, Van Tuyl and the second author for lines in P 3 , we develop asymptotic upper bounds for the least degree of a homogeneous form vanishing to order at least m on a union of disjoint r dimensional planes in P n for n ≥ 2r + 1. These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in P 2 is not an exotic statement but rather a manifestation of a much more general phenomenon which seems to have been overlooked so far.

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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 Nagatatype 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.