Linear systems on a special rational surface

We study the Hilbert series of two families of ideals generated by powers of linear forms in $\mathbb{K}[x_1,x_2,x_3]$. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints in $\mathbb{P}^{2}$. This is equivalent to studying the dimension of a linear system on a blow up of $\mathbb{P}^2$. 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 $K^2 > 0$ 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 \cite{ps}.

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Published 1 January 2004