We study the Hilbert series of two families of ideals generated by powers of linear forms in K [x1, x2, x3]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints in P2. This is equivalent to studying the dimension of a linear system on a blow up of P2. We determine the classes of the negative curves, then apply an algorithm of Harbourne to reduce to an effective, nef divisor. Combining Harbourne's results on rational surfaces with K2 > 0 and Riemann-Roch yields a formula for the Hilbert series. For one family of ideals, this proves the n = 3 case of a conjecture posed by Postnikov and Shapiro "as a challenge to the commutative algebra community" (after this proof was communicated to them, they found a combinatorial proof for all n). For the second family of ideals, it yields a formula which Postnikov and Shapiro were unable to obtain via combinatorial methods. We conjecture a formula for the minimal free resolution of one family of ideals, and show that a member of the second family of ideals provides a counterexample to a conjecture made by Postnikov and Shapiro in .
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