Multigraded regularity of complete intersections

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 $\text{codim}(V)$. 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.