Linear subspaces, symbolic powers and Nagata type conjectures
  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. 
Marcin Dumnicki, Brian Harbourne, Tomasz Szemberg, Halszka   Tutaj-Gasi\'nska
math.AG