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 J_{phi} generated by powers of linear
forms in k[x_{1},...,x_{n}]. Using the
results of EmsalemIarrobino, we formulate this as a question
about fatpoints in P^{n1}. In the three variable
case this is equivalent to studying the dimension of a linear
series 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, n.e.f. divisor and using RiemannRoch 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 J_{phi} 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.
