ACM sets of points in multiprojective space

Abstract

If X is a finite set of points in a multiprojective space P n 1 ×· · ·×P nr with r ≥ 2, then X may or may not be arithmetically Cohen-Macaulay (ACM). For sets of points in P 1 × 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 × 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.

Separators of points in a multiprojective space

“…For any finite set of points X ⊆ P n 1 × · · · × P nr , it can be shown (see, for example [12, Theorem 2.1]) that dim R/I X = r and 1 ≤ depth R/I X ≤ r. When depth R/I X = r, then we say X is arithmetically Cohen-Macaulay (ACM). Although it remains an open problem to classify ACM sets of points in a multiprojective space (see [12] for some work on this problem), it can be shown that the separators of ACM sets of points have a particularly nice property:…”

“…Although the converse of Theorem 2.6 fails to hold in general (see [12,Example 5.10]), the converse holds in P 1 × P 1 as first demonstrated by Marino:…”

“…Now deg X (Q 2,2 )+ (2, 2) = (4, 4) is a shift that appears in the minimal multigraded resolution of I X . However, Y is not ACM because Y is the nonACM set of points of [12,Example 3.3].…”

“…Marino showed that X ⊆ P 1 × P 1 is arithmetically Cohen-Macaulay (ACM) if and only if for every P ∈ X, | deg X (P)| = 1. More recently, the authors [12] extended some of Marino's results to an arbitrary multiprojective space; in particular, if X ⊆ P n 1 × · · · × P nr and is ACM, then every point P ∈ X has | deg X (P)| = 1, but the converse no longer holds.…”

“…While a cursory introduction to the properties of separators appears in [12], in this paper we wish to provide a more systematic introduction, thereby extending our understanding of points in a multiprojective space (see, for example, [7,8,9,10,11,20,21,22], for more on these points). In Section 2, we relate the Hilbert functions of X and X \ {P} using the set deg X (P) (see Theorem 2.2), thus introducing a multigraded analog of a result of Geramita, Maroscia, and Roberts [6].…”