On the arithmetically Cohen-Macaulay property for sets of points in multiprojective spaces

Published version: We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially $(\mathbb P^1)^n$\mathrm{.}Acombinatorialcharacterization,the$(\star)$-property,isknownin$\mathbb P^1 \times \mathbb P^1$\mathrm{.}Weproposeacombinatorialproperty,$(\star_n)$,thatdirectlygeneralizesthe$(\star)$-propertyto$(\mathbb P^1)^n$forlarger$n$\mathrm{.}Weshowthat$X$isACMifandonlyifitsatisfiesthe$(\star_n)$-property. The main tool for several of our results is an extension to the multiprojective setting of certain liaison methods in projective space. Corrigendum: We correct a mistake in the cited paper. It introduced a combinatorial property, the $({\star }_n)$-property, for a finite set of points $X$ in $({\mathbb{P}}^1{)}^n$ and claimed that this property holds if and only if $X$ is ACM. In fact $X$ being ACM is a sufficient condition for the $({\star }_n)$-property, but we only prove that it is necessary when $n=3$, and we give a counterexample when $n=4$.