On the Hilbert function of a finite scheme contained in a quadric surface

Abstract

Let X be a zero-dimensional scheme contained in a multiprojective space. Let s i be the length of the projection of X onto the i-th component of the multiprojective space. A result of A. Van Tuyl states that the Hilbert function of X, in the case when X is reduced, is completely determined by its restriction to the product of the intervals [0, s i − 1]. We prove that the same is also true for non-reduced schemes X.