ar X iv : m at h / 02 03 07 1 v 2 [ m at h . A C ] 4 N ov 2 00 2 FAT POINTS IN P 1 × P 1 AND THEIR HILBERT FUNCTIONS

We study the Hilbert functions of fat points in P 1 × P 1. If Z ⊆ P 1 × P 1 is an arbitrary fat point scheme, then it can be shown that for every i and j the values of the Hilbert function H Z (l, j) and H Z (i, l) eventually become constant for l ≫ 0. We show how to determine these eventual values by using only the multiplicities of the points, and the relative positions of the points in P 1 × P 1. This enables us to compute all but a finite number values of H Z without using the coordinates of points. We also characterize the ACM fat points schemes using our description of the eventual behaviour. In fact, in the case that Z ⊆ P 1 × P 1 is ACM, then the entire Hilbert function and its minimal free resolution depend solely on knowing the eventual values of the Hilbert function.