Resolutions of Ideals of Uniform Fat Point Subschemes of P²

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Let I be the ideal corresponding to a set of general points p₁,...,pₙ ∈ P². 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.