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Speaker: Hal Schenck, Texas A&M

Title: Linear series on a special rational surface

Abstract: We study the Hilbert series of a family of ideals Jphi generated by powers of linear forms in k[x1,...,xn]. Using the results of Emsalem-Iarrobino, we formulate this as a question about fatpoints in Pn-1. In the three variable case this is equivalent to studying the dimension of a linear series on a blow up of P2. 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, n.e.f. divisor and using Riemann-Roch yields a formula for the Hilbert series. This proves the n=3 case of a theorem, which Postnikov and Shapiro later showed true for all n. Postnikov and Shapiro observe that for a family of ideals closely related to Jphi a similar formula often seems to hold, although counterexamples exist for n=4 and n=5. Our methods allow us to prove that for n=3 the formula is indeed true.

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