  This paper is concerned with determining the number of generators in each
degree for minimal sets of homogeneous generators for saturated ideals defining
fat point subschemes $Z=m_1p_1+ ... +m_rp_r$ for general sets of points $p_i$
of $P^2$. For thin points (i.e., m_i=1 for all i), a solution is known, in
terms of a maximal rank property. Although this property in general fails for
fat points, we show it holds in an appropriate asymptotic sense. In the uniform
(i.e., $m_1= ... =m_r$) case, we determine all failures of this maximal rank
property for $r\le 9$, and we develop evidence for the conjecture that no other
failures occur for $r > 9$.
