  We investigate the expected dimensionality of linear systems with general fat
points on certain surfaces using an approach by specialization to elliptic
surfaces. For the projectivization of the Atiyah bundle over an elliptic curve
with a certain polarization, we observe that the special case of only one fat
point implies the general case of arbitrarily many fat points, as well as
results concerning other surfaces. We conjecture that this special case holds
in characteristic 0, but prove that it fails in any positive characteristic.
