×

zbMATH — the first resource for mathematics

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
Software:
CoCoA
PDF BibTeX XML Cite
Full Text: DOI EuDML arXiv
References:
[1] S. Abrescia, L. Bazzotti, and L. Marino, Conductor degree and socle degree,Matematiche (Catania) 56 (2003), 129–148. · Zbl 1172.13306
[2] L. Bazzotti and M. Casanellas, Separators of points on algebraic surfaces,J. Pure Appl. Algebra 207 (2006), 319–326. · Zbl 1107.14038 · doi:10.1016/j.jpaa.2005.10.016
[3] L. Bazzotti, Sets of points and their conductor,J. Algebra 283 (2005), 799–820. · Zbl 1119.13011 · doi:10.1016/j.jalgebra.2004.09.031
[4] J. Chan, C. Cumming, and H.T. Hà, CohenMacaulay multigraded modules, (preprint 2007), arXiv:0705.1839, to appear inIllinois Journal of Mathematics.
[5] CoCoATeam, CoCoA: a system for doing Computations in Commutative Algebra, Available at http://cocoa.dima.unige.it
[6] A.V. Geramita, P. Maroscia, and L. Roberts, The Hilbert function of a reducedk-algebra,J. London Math. Soc. (2)28 (1983), 443–452. · Zbl 0535.13012 · doi:10.1112/jlms/s2-28.3.443
[7] S. Giuffrida, R. Maggioni, and A. Ragusa, On the postulation of 0-dimensional subschemes on a smooth quadric,Pacific J. Math. 155 (1992), 251–282. · Zbl 0757.14027 · doi:10.2140/pjm.1992.155.251
[8] S. Giuffrida, R. Maggioni, and A. Ragusa, Resolutions of 0-dimensional subschemes of a smooth quadric,Zero-dimensional schemes (Ravello, 1992), 191–204, de Gruyter, Berlin, 1994. · Zbl 0826.14029
[9] S. Giuffrida, R. Maggioni, and A. Ragusa, Resolutions of generic points lying on a smooth quadric,Manuscripta Math. 91 (1996), 421–444. · Zbl 0873.14041 · doi:10.1007/BF02567964
[10] E. Guardo,Schemi di ”Fat Points”, Ph.D. thesis, Università di Messina, 2000.
[11] E. Guardo, Fat points schemes on a smooth quadric,J. Pure Appl. Algebra 162 (2001), 183–208. · Zbl 1044.14025 · doi:10.1016/S0022-4049(00)00123-7
[12] E. Guardo, A survey on fat points on a smooth quadric,Algebraic structures and their representations, 61–87, Contemp. Math.376, Amer. Math. Soc., Providence, RI, 2005. · Zbl 1087.14007
[13] H.T. Hà and A. Van Tuyl, The regularity of points in multiprojective spaces,J. Pure Appl. Algebra 187 (2004), 153–167. · Zbl 1036.14024 · doi:10.1016/j.jpaa.2003.07.006
[14] L. Marino, Conductor and separating degrees for sets of points in \(\mathbb{P}\)r and in \(\mathbb{P}\)1 \(\times\) \(\mathbb{P}\)1,Boll. Unione Mat. Ital. Sez. B Artic. Ric. Mat. (8)9 (2006), 397–421. · Zbl 1178.13007
[15] L. Marino, The minimum degree of a surface that passes through all the points of a 0dimensional scheme but a pointP, Algebraic structures and their representations, 315–332, Contemp. Math.376, Amer. Math. Soc., Providence, RI, 2005. · Zbl 1084.14009
[16] L. Marino,A characterization of ACM 0-dimensional subschemes of \(\mathbb{P}\)1 \(\times\) \(\mathbb{P}\)1, to appear.
[17] F. Orecchia, Points in generic position and conductors of curves with ordinary singularities,J. London Math. Soc. (2)24 (1981), 85–96. · Zbl 0492.14017 · doi:10.1112/jlms/s2-24.1.85
[18] J. Sidman and A. Van Tuyl, Multigraded regularity: syzygies and fat points,Beiträge Algebra Geom.47 (2006), 67–87. · Zbl 1095.13012
[19] A. Van Tuyl,Sets of points in multiprojective spaces and their Hilbert function, Ph.D. thesis, Queen’s University, 2001.
[20] A. Van Tuyl, The border of the Hilbert function of a set of points in $$\(\backslash\)mathbb{P}\^{n_1 } \(\backslash\)times \(\backslash\)cdots \(\backslash\)times \(\backslash\)mathbb{P}\^{n_k } $$ ,J. Pure Appl. Algebra 176 (2002), 223–247. · Zbl 1019.13008 · doi:10.1016/S0022-4049(02)00072-5
[21] A. Van Tuyl, The Hilbert functions of ACM sets of points in $$\(\backslash\)mathbb{P}\^{n_1 } \(\backslash\)times \(\backslash\)cdots \(\backslash\)times \(\backslash\)mathbb{P}\^{n_k } $$ ,J. Algebra 264 (2003), 420–441. · Zbl 1039.13008 · doi:10.1016/S0021-8693(03)00232-1
[22] A. Van Tuyl, The defining ideal of a set of points in multiprojective space,J. London Math. Soc. (2)72 (2005), 73–90. · Zbl 1086.13005 · doi:10.1112/S0024610705006459
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.