# zbMATH — the first resource for mathematics

Fat points schemes on a smooth quadric. (English) Zbl 1044.14025
Summary: We study zero-dimensional fat points schemes on a smooth quadric $$Q\cong\mathbb{P}^1\times\mathbb{P}^1$$, and we characterize those schemes which are arithmetically Cohen-Macaulay (aCM for short) as subschemes of $$Q$$ giving their Hilbert matrix and bigraded Betti numbers. In particular, we can compute the Hilbert matrix and the bigraded Betti numbers for fat points schemes with homogeneous multiplicities and whose support is a complete intersection. Moreover, we find a minimal set of generators for schemes of double points whose support is a CM.

##### MSC:
 14M05 Varieties defined by ring conditions (factorial, Cohen-Macaulay, seminormal) 14A15 Schemes and morphisms 14M10 Complete intersections 13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.) 13D02 Syzygies, resolutions, complexes and commutative rings 13C40 Linkage, complete intersections and determinantal ideals
Full Text:
##### References:
 [1] Alexander, J., Singularités imposables en position générale aux hypersurfaces de $$P\^{}\{n\}$$, Comput. math., 68, 305-354, (1988) · Zbl 0675.14025 [2] Alexander, J.; Hirschowitz, A., Une lemme d’horace différentielle: application aux singularités hyperquartiques de $$P\^{}\{5\}$$, J. algebra geom., 1, 411-426, (1992) · Zbl 0784.14001 [3] Alexander, J.; Hirschowitz, A., La méthode d’horace éclatée: application á l’interpolation en degré quatre, Invent. math., 107, 585-602, (1992) · Zbl 0784.14002 [4] Alexander, J.; Hirschowitz, A., Polynomial interpolation in several variables, J. algebra geom., 4, 201-222, (1995) · Zbl 0829.14002 [5] Catalisano, M.V., Fat points on a conic, Comm. algebra, 19, 2153-2168, (1991) · Zbl 0758.14005 [6] Catalisano, M.V.; Gimigliano, A., On the Hilbert function of fat points on a rational normal cubic, J. algebra, 183, 245-265, (1996) · Zbl 0863.14028 [7] C. Ciliberto, R. Miranda, On the dimension of linear systems of plane curves with general multiple base points, preprint, 1997. [8] E.D. Davis, A.V. Geramita, The Hilbert function of a special class of 1-dimensional Cohen-Macaulay graded algebras, The Curves Seminar at Queen’s, vol. III, Queen’s Paper in Pure and Applied Mathematics, No. 67, 1984. [9] S. Fitchett, B. Harbourne, S. Holay, Resolutions of ideals of uniform fat point subschemes of $$P\^{}\{2\}$$, Preprint available at http://www.math.unl.edu/∼bharbour/, 2000. · Zbl 1028.14020 [10] A. Gimigliano, Our thin knowledge of fat points, The Curves Seminar at Queen’s, vol. VI, Queen’s Paper in Pure and Applied Mathemtics, No. 83, 1989, pp. B1-B50. [11] S. Giuffrida, Hilbert Function of a 0-cycle in $$P\^{}\{2\}$$, Le Mat. XL (1985) I-II. [12] Giuffrida, S.; Maggioni, R.; Ragusa, A., On the postulation of 0-dimensional subschemes on a smooth quadric, Pacific J. math., 155, 2, 251-282, (1992) · Zbl 0723.14035 [13] S. Giuffrida, R. Maggioni, A. Ragusa, Resolutions of zero-dimensional subschemes of a smooth quadric, Proceedings of the International Conference held in Ravello, June 8-13, 1992, pp. 191-204. · Zbl 0826.14029 [14] Giuffrida, S.; Maggioni, R.; Ragusa, A., Resolutions of generic points lying on a smooth quadric, Manuscripta math., 91, 421-444, (1996) · Zbl 0873.14041 [15] Harbourne, B., Complete linear systems on rational surfaces, Trans. AMS, 289, 213-226, (1985) · Zbl 0609.14004 [16] Harbourne, B., Blowings up of $$P\^{}\{2\}$$ and their blowings down, Duke math. J., 52, 129-148, (1985) · Zbl 0577.14025 [17] B. Harbourne, The geometry of rational surfaces and Hilbert functions of points in the plane, Canadian Mathematical Society Conference Proceedings, vol. 6, 1986, pp. 95-111. · Zbl 0611.14002 [18] B. Harbourne, Points in good position in $$P\^{}\{2\}$$, in: Zero-Dimensional Schemes, Proceedings of International Conference held in Ravello, Italy, June 8-13 June 1992, De Gruyter, Berlin, 1994. [19] Harbourne, B., Rational surfaces with K2>0, Proc. amer. math. soc., 124, 727-733, (1996) · Zbl 0874.14025 [20] B. Harbourne, Anticanonical rational surfaces, Trans. Amer. Math. Soc. 1997. · Zbl 0860.14006 [21] Harbourne, B., Free resolutions of fat point ideals on $$P\^{}\{2\}$$, J. pure appl. algebra, 125, 213-234, (1998) · Zbl 0901.14030 [22] Harbourne, B., The ideal generation problem for fat points, J. pure appl. algebra, 145, 165-182, (2000) · Zbl 0942.14024 [23] Hirschowitz, A., Une conjecture pour la cohomologie des diviseurs sur LES surfaces rationelles génériques, J. reine angew. math., 397, 208-213, (1989) · Zbl 0686.14013 [24] A. Iarrobino, Inverse systems of a symbolic power III: thin algebras and fat points, 1994. · Zbl 0899.13016
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.