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

TL;DR

This paper investigates the ACM property of unions of lines in b^1 b^2, exploring conditions for ACM status, modifications to achieve ACM, and combinatorial characterizations, with applications to b^3.

Contribution

It provides new criteria and combinatorial characterizations for when unions of lines in b^1 b^2 are ACM and how to modify non-ACM configurations to become ACM.

Findings

Characterization of ACM unions of lines in b^1 b^2

Conditions for 'fattening' lines to achieve ACM

Introduction of fully v-connected configurations

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 $\mathbb P^3$.