Multigraded regularity of complete intersections

TL;DR

This paper studies the multigraded regularity of complete intersection schemes in multiprojective spaces, providing explicit Hilbert function computations and establishing a sharp upper bound for their multigraded regularity.

Contribution

It offers explicit Hilbert function values and a precise upper bound for the multigraded regularity of 0-dimensional complete intersections in multiprojective spaces.

Findings

Hilbert functions depend only on dimensions and generator bidegrees.

Explicit computations for 0-dimensional complete intersections.

Established a sharp upper bound for multigraded regularity.

Abstract

$V$ is a complete intersection scheme in a multiprojective space if it can be defined by an ideal $I$ with as many generators as $\textrm{codim}(V)$. We investigate the multigraded regularity of complete intersections scheme in $\mathbb{P}^n\times \mathbb{P}^m$. We explicitly compute many values of the Hilbert functions of $0$-dimensional complete intersections. We show that these values only depend upon $n,m$, and the bidegrees of the generators of $I$. As a result, we provide a sharp upper bound for the multigraded regularity of $0$-dimensional complete intersections.