Linear systems on a special rational surface

Abstract

We study the Hilbert series of a family of ideals J_ϕgenerated by powers of linear forms in k[x_1,...,x_n]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints in P^{n-1}. In the three variable case this is equivalent to studying the dimension of a linear system on a blow up of P^2. The ideals that arise have the points in very special position, but because there are only seven points, we can apply results of Harbourne to obtain the classes of the negative curves. Reducing to an effective, nef divisor and using Riemann-Roch yields a formula for the Hilbert series. This proves the n=3 case of a conjecture of Postnikov and Shapiro, which they later showed true for all n. Postnikov and Shapiro observe that for a family of ideals closely related to J_ϕa similar result often seems to hold, although counterexamples exist for n=4 and n=5. Our methods allow us to prove that for n=3 an analogous formula is indeed true. We close with a counterexample to a conjecture Postnikov and Shapiro make about the minimal free resolution of these ideals.