Linear subspaces, symbolic powers and Nagata type conjectures

Inspired by results of Guardo, Van Tuyl and the second author for lines in P3P3, 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 PnPn for n⩾2r+1n⩾2r+1. These considerations lead to new conjectures that suggest that the well known conjecture of Nagata for points in P2P2 is not an exotic statement but rather a manifestation of a much more general phenomenon which seems to have been overlooked so far.