FAT POINTS IN P 1 × P 1 AND THEIR HILBERT FUNCTIONS

Abstract

We study the Hilbert functions of fat points in P 1 × P 1. We associate to an arbitrary scheme of fat points Z two tuples of non-negative integers αZ and βZ that depend only upon the multiplicities and relative positions of the points. We then show that all but a finite number of values of the Hilbert function of Z can be computed directly from αZ and βZ. We also characterize the ACM schemes Z using only the combinatorial information contained in αZ and βZ. In the case that Z is ACM, then the entire Hilbert function of Z and its minimal free resolution depend only upon αZ and βZ.