The ACM property for unions of lines in $\mathbb P^1 \times \mathbb P^2$

Abstract: This paper examines the Arithmetically Cohen-Macaulay (ACM) property for certain codimension 2 varieties in $\mathbb P^1\times \mathbb P^2$ called sets of lines in $\mathbb P^1\times \mathbb P^2$ (not necessarily reduced). We discuss some obstacles to finding a general characterization. We then consider certain classes of such curves, and we address two questions. First, when are they themselves ACM? Second, in a non-ACM reduced configuration, is it possible to replace one component of a primary (prime) decomposition by a suitable power (i.e. to "fatten" one line) to make the resulting scheme ACM? Finally, for our classes of such curves, we characterize the locally Cohen-Macaulay property in combinatorial terms by introducing the definition of a fully v-connected configuration. We apply some of our results to give analogous ACM results for sets of lines in $\mathbb P^3$.