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 **P**^{n}. It is natural to try to generalize Macaulay's Theorem to non-reduced schemes. Given a set of points Y = {P_{1}, P_{2}, ..., P_{k}} in **P**^{n} and non-negative integers m_{1}, m_{2}, ..., m_{k}, the intersection I of the ideals
I_{P1}^{m1}, I_{P2}^{m2}, ..., I_{Pk}^{mk},
where I_{Pi} is the ideal generated by all forms that vanish at the point P_{i}, defines a subscheme in **P**^{n} 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 m_{i} as the points P_{i} 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 m_{i} (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 P_{i}. 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.