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ACM sets of points in multiprojective space. (English) Zbl 1146.13012
Let $${\mathbb X}$$ be a finite set of points in a multiprojective space $${\mathbb P}^{n_1}\times ... \times {\mathbb P}^{n_r}$$. $${\mathbb X}$$ is arithmetically Cohen-Macaulay (a.C.M.) if its coordinate ring $$R/I_ {\mathbb X}$$ is C.M. (i.e. depth $$R/I_ {\mathbb X} = \dim R/I_ {\mathbb X}$$).
When $$r=1$$, $${\mathbb X}$$ is always a.C.M., but for $$r\geq 2$$ it is an open problem how to geometrically determine if $${\mathbb X}$$ is a.C.M. or not, and to classify a.C.M sets of points.
The problem has been widely studied for $$r=2$$, $$n_1=n_2=1$$ (i.e. on a smooth quadric surface), and in this case a.C.M. sets can be classified via their Hilbert function, moreover we have that $${\mathbb X}\subset {\mathbb P}^{1}\times{\mathbb P}^{1}$$ is a.C.M. if and only if whenever it contains $$P_1\times Q_1$$ and $$P_2\times Q_2$$, then either $$P_1\times Q_2$$ or $$P_2\times Q_1$$ are in it too (property $$\star$$).
In this paper the authors show that if $${\mathbb X}\subset {\mathbb P}^{1}\times{\mathbb P}^{n}$$ and is a.C.M. then property $$\star$$ holds, but the converse in false (unless $$n=1$$), and give counterexample for $$n=2$$. They also prove that we cannot expect that a.C.M. sets of points $${\mathbb X}\subset {\mathbb P}^{n_1}\times...\times{\mathbb P}^{n_r}$$ can be classified via their Hilbert function, in general.
In the last section the authors study relations between $${\mathbb X}$$ being a.C.M. and the (multi)degree of the separators of its points, where a separator of $$P\in {\mathbb X}$$ is a form $$F\in R$$ such that $$F(P)=0$$ and $$F(Q)\neq 0$$, $$\forall Q\in {\mathbb X}-\{P\}$$, namely they generalize the result (known on the quadric surface) that if $${\mathbb X}$$ is a.C.M. then $$\forall P\in {\mathbb X}$$ the minimal degree of separators of $$P$$ is unique. Moreover, they also show that the converse does not hold in general.

##### MSC:
 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series 13D02 Syzygies, resolutions, complexes and commutative rings
##### Keywords:
Cohen-Macaulay; multiprojective space; separator
CoCoA
Full Text:
##### References:
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