The Relevant Domain of the Hilbert Function of a Finite Multiprojective Scheme

TL;DR

This paper extends Van Tuyl's result by proving that the Hilbert function of any zero-dimensional scheme in a multiprojective space, whether reduced or not, is determined by its restriction to a specific product of intervals.

Contribution

It generalizes Van Tuyl's theorem to include non-reduced schemes, showing the Hilbert function is determined by a restricted subset.

Findings

Hilbert function determined by restriction to product of intervals

Extension of Van Tuyl's result to non-reduced schemes

Applicable to zero-dimensional schemes in multiprojective spaces

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 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.