Special arrangements of lines: codimension two ACM varieties in ${\mathbb{P}}^1\times {\mathbb{P}}^1\times {\mathbb{P}}^1$

In this paper we investigate special arrangements of lines in multiprojective spaces. In particular, we characterize codimensional two arithmetically Cohen-Macaulay (ACM) varieties in $\mathbb P^1\times\mathbb P^1\times\mathbb P^1$, called varieties of lines. We also describe their ACM property from combinatorial algebra point of view.