Splittings of Ideals of Points in $\mathbb{P}^{1}\times\mathbb{P}^{1}$

Abstract

Let $I_\mathbb{X}$ be the bihomogeneous ideal of a finite set of points $\mathbb{X} \subseteq \mathbb{P}^1 \times \mathbb{P}^1$. The purpose of this note is to consider ``splittings'' of the ideal $I_\mathbb{X}$, that is, finding ideals $J$ and $K$ such that $I_\mathbb{X} = J+K$, where $J$ and $K$ have prescribed algebraic or geometric properties. We show that for any set of points $\mathbb{X}$, we cannot partition the generators of $I_\mathbb{X}$ into two ideals of points. The best case scenario is where at most one of $J$ or $K$ is an ideal of points. To remedy this we introduce the notion of unions of lines and ACM (Arithmetically Cohen-Macaulay) points which allows us to say more about splittings. For a set $\mathbb{W}$ of unions of lines and ACM sets of points, we can write $I_\mathbb{W} = J + K$ where both $J$ and $K$ are ideals of unions of lines and ACM points as well. When $\mathbb{W}$ is a union of lines and ACM points, we discuss some consequences for the graded Betti numbers of $I_{\mathbb{W}}$ in terms of these splittings.