Classification of ACM sets of points in $\mathbb{P}^{1} \times \mathbb{P}^{1}$

Abstract

The coordinate ring of a finite sets of points in $\mathbb{P}^{n}$ is always a Cohen-Macaulay ring. However, the multigraded coordinate ring of a set of points in $\mathbb{P}^{1} \times \mathbb{P}^{1}$, or more generally, a set of points in $\mathbb{P}^{n_{1}} \times \cdots \times \mathbb{P}^{n_{r}}$, may fail to have this highly desirable property. This feature is one of the fundamental differences between sets of points in a single projective space and sets of points in a multiprojective space.