  Let I be the ideal corresponding to a set of general points $p_1,...,p_n \in
P^2$. There recently has been progress in showing that a naive lower bound for
the Hilbert functions of symbolic powers $I^{(m)}$ is in fact attained when
n>9. Here, for m sufficiently large, the minimal free graded resolution of
$I^{(m)}$ is determined when n>9 is an even square, assuming only that this
lower bound on the Hilbert function is attained. Under ostensibly stronger
conditions (that are nonetheless expected always to hold), a similar result is
shown to hold for odd squares, and for infinitely many m for each nonsquare n
bigger than 9. All results hold for an arbitrary algebraically closed field k.
