On the relevant domain of the Hilbert function of a finite multiprojective scheme

Abstract Let X be a zero-dimensional reduced subscheme of a multiprojective space $\mathbb {V} $ . Let $s_i$ be the length of the projection of X onto the i th component of $\mathbb {V}$ . A result of Van Tuyl states that the Hilbert function of X is completely determined by its restriction to the product of the intervals $[0, s_i - 1]$ . We extend this result to arbitrary zero-dimensional subschemes of $\mathbb {V}$ .