The Hilbert functions of ACM sets of points in P^{n_1} x ... x P^{n_k}

The Hilbert functions of sets of distinct points in P^n have been characterized. We show that if we restrict to sets of distinct of points in P^{n_1} x ... x P^{n_k} that are also arithmetically Cohen-Macaulay (ACM for short), then there is a natural generalization of this result. We begin by determining the possible values for the invariants K-dim R/Ix and depth R/Ix, where R/Ix is the coordinate ring associated to a set of distinct points X in P^{n_1} x ... x P^{n_k}. At the end of this paper we give a new characterization of ACM sets of points in P^1 x P^1.Comment: 18 pages, v.2 to appear in J. Alg. Sections 2 and 3 have been combined, Section 5 has been removed, and Section 6 has been split into two section

ACM sets of points in multiprojective space

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.Comment: 21 pages; revised final version; minor corrections; to appear in Collectanea Mathematic

Simplicial complexes and Macaulay's inverse systems

Let $\Delta$ be a simplicial complex on $V = \{x_1,...,x_n\}$, with Stanley-Reisner ideal $I_{\Delta}\subseteq R = k[x_1,...,x_n]$. The goal of this paper is to investigate the class of artinian algebras $A=A(\Delta,a_1,...,a_n)= R/(I_{\Delta},x_1^{a_1},...,x_n^{a_n})$, where each $a_i \geq 2$. By utilizing the technique of Macaulay's inverse systems, we can explicitly describe the socle of $A$ in terms of $\Delta$. As a consequence, we determine the simplicial complexes, that we will call {\em levelable}, for which there exists a tuple $(a_1,...,a_n)$ such that $A(\Delta,a_1,...,a_n)$ is a level algebra.Comment: Very minor changes. To appear in Math.