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Let Z be a fat point scheme in P^2 supported on general points. Here we provethat if the multiplicities are at most 3 and the length of Z is sufficientlyhigh then the number of generators of the homogeneous ideal I_Z in each degreeis as small as numerically possible. Since it is known that Z has maximalHilbert function, this implies that Z has the expected minimal free resolution.

On the minimal free resolution for fat point schemes of multiplicity at most 3 in P^2