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.
Categories: math.AC math.AG