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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.

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
Full Text: DOI EuDML arXiv
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