Speaker: Susan Cooper

Title: Bounding Hilbert Functions of Fat Points

Abstract: A famous theorem due to Macaulay characterizes the Hilbert functions of finite sets of distinct, reduced points in Pn. It is natural to try to generalize Macaulay's Theorem to non-reduced schemes. Given a set of points Y = {P1, P2, ..., Pk} in Pn and non-negative integers m1, m2, ..., mk, the intersection I of the ideals IP1m1, IP2m2, ..., IPkmk, where IPi is the ideal generated by all forms that vanish at the point Pi, defines a subscheme in Pn which is known as a fat point scheme.

In general, it is not yet known what the Hilbert functions are for I with fixed multiplicities mi as the points Pi vary. However, if n = 2 and the number of points is 8 or less, we can write down all of the possible Hilbert functions for any given set of multiplicities mi (due to Guardo-Harbourne and Geramita-Harbourne-Migliore).

In this talk we focus on what we can say, when n = 2, given information about what collinearities occur among the points Pi. Using this information and an argument motivated by Bézout's Theorem, we obtain upper and lower bounds for the Hilbert function of I. Moreover, we give a simple criterion for when the bounds coincide yielding a precise calculation of the Hilbert function in this case.

This is joint work with B. Harbourne and Z. Teitler.