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 ${\mathbb{P}}^3$.

本文研究了某些Codimension 2变种的算术Cohen-Macaulay（ACM）性质${\mathbb{P}}^{\mathrm{1个}}\times {\mathbb{P}}^{\mathrm{2个}}$所谓的套系中${\mathbb{P}}^{\mathrm{1个}}\times {\mathbb{P}}^{\mathrm{2个}}$（不一定减少）。我们讨论了寻找一般特征的一些障碍。然后，我们考虑此类曲线的某些类别，并解决两个问题。首先，他们自己何时是ACM？第二，在非ACM简化配置中，是否可以用合适的功率（即“加肥”一条线路）替换一次（原始）分解的一个分量，以形成最终的方案ACM？最后，对于此类曲线的类别，我们通过引入完全v型连接的配置的定义，以组合术语来表征局部Cohen-Macaulay属性。我们应用一些结果来为A中的行集提供类似的ACM结果${\mathbb{P}}^3$。