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

TL;DR

This paper characterizes when finite sets of points in multiprojective spaces, especially $(b P^1)^n$, are arithmetically Cohen-Macaulay using a new combinatorial property called $(igstar_n)$, extending previous results from $(b P^1)^1$.

Contribution

It introduces the $(igstar_n)$-property as a combinatorial criterion for ACM sets in $(b P^1)^n$, generalizing known characterizations and providing new liaison-based tools.

Findings

$(igstar_n)$-property characterizes ACM sets for $n=3$.

Counterexample shows the property is not sufficient for $n=4$.

Extension of liaison methods to 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 $n=4$.