  Prompted by results of Guardo, Van Tuyl and the second author for lines in
projective 3 space, 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 projective n space for n at least 2r+1. These
considerations lead to new conjectures that suggest that the well known
conjecture of Nagata for points in the projective plane is not sporadic, but
rather a special case of a more general phenomenon.
