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

Abstract

Published version: We study the arithmetically Cohen-Macaulay (ACM) property for finite sets of points in multiprojective spaces, especially $({\mathbb{P}}^1{)}^n$. A combinatorial characterization, the $(\star )$-property, is known in ${\mathbb{P}}^1\times {\mathbb{P}}^1$. We propose a combinatorial property, $({\star }_n)$, that directly generalizes the $(\star )$-property to $({\mathbb{P}}^1{)}^n$ for larger $n$. We show that $X$ is ACM if and only if it satisfies the $({\star }_n)$-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 $n=4$.

This is a corrigendum of the paper 1702.01199v2, which appeared in the Proceedings of the AMS in 2018. It contains only the correction of Theorem 3.16 and a counterexample to the original more general statement pointed out to us by G. Fløystad. It should be read in conjunction with the latest posted version of the full paper for context