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This paper examines the Arithmetically Cohen-Macaulay (ACM) property forcertain codimension 2 varieties in $\mathbb P^1\times \mathbb P^2$ called setsof lines in $\mathbb P^1\times \mathbb P^2$ (not necessarily reduced). Wediscuss some obstacles to finding a general characterization. We then considercertain classes of such curves, and we address two questions. First, when arethey themselves ACM? Second, in a non-ACM reduced configuration, is it possibleto 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 ourclasses of such curves, we characterize the locally Cohen-Macaulay property incombinatorial terms by introducing the definition of a fully v-connectedconfiguration. We apply some of our results to give analogous ACM results forsets of lines in $\mathbb P^3$.

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