ACM sets of points in multiprojective space

TL;DR

This paper explores the properties and classifications of finite point sets in multiprojective spaces, revealing limitations of existing classifications and proposing new conditions for arithmetically Cohen-Macaulay sets.

Contribution

It generalizes the study of ACM point sets from P^1 x P^1 to arbitrary multiprojective spaces and introduces new necessary and sufficient conditions.

Findings

Existing classifications for ACM points in P^1 x P^1 do not extend to higher multiprojective spaces

New conditions for a set of points to be ACM are proposed

Some classifications are shown to fail in the general case

Abstract

If X is a finite set of points in a multiprojective space P^n1 x ... x P^nr with r >= 2, then X may or may not be arithmetically Cohen-Macaulay (ACM). For sets of points in P^1 x P^1 there are several classifications of the ACM sets of points. In this paper we investigate the natural generalizations of these classifications to an arbitrary multiprojective space. We show that each classification for ACM points in P^1 x P^1 fails to extend to the general case. We also give some new necessary and sufficient conditions for a set of points to be ACM.