Special arrangements of lines: codimension two ACM varieties in $mathbb P^1timesmathbb P^1timesmathbb P^1$

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