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Families of Artinian and one-dimensional algebras. (English) Zbl 1129.14009
This paper focuses on families of Artinian or one-dimensional quotients of a polynomial ring \(R\). Let \(H\) be a Hilbert function and let GradAlg\(^H (R)\) be the scheme parametrizing all graded quotients of \(R\) with Hilbert function \(H\). Let \(B \rightarrow A\) be a graded surjection of quotients of \(R\), with Hilbert functions \(H_B\) and \(H_A\) respectively. If \(\dim A = 0\) or 1 and making some additional assumptions on both \(A\) and \(B\), the author gives close connections between GradAlg\(^{H_B}(R)\) and GradAlg\(^{H_A}(R)\). These connections involve for instance smoothness and dimension of these parameter schemes. In a more general setting he describes the dual of the tangent and obstruction space of graded deformations. He then applies this machinery to the case of level algebras of Cohen-Macaulay type 2 (this is a natural first case after the Gorenstein algebras). As a result, he proves a conjecture of Iarrobino on the existence of at least two irreducible components of GradAlg\(^H(R)\) when \(H = (1,3,6,10,14,10,6,2)\), such that the general elements of these components are level. The work is very technical, but many examples are given to illustrate the methods. Similar parameter schemes have been studied in the past, but generally those papers have considered the reduced scheme structure, while GradAlg\(^H (R)\) may be non-reduced.

MSC:
14C05 Parametrization (Chow and Hilbert schemes)
13E10 Commutative Artinian rings and modules, finite-dimensional algebras
13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D10 Deformations and infinitesimal methods in commutative ring theory
13C40 Linkage, complete intersections and determinantal ideals
Software:
Macaulay2
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References:
[1] André, M., Méthode simpliciale en algèbre homologique et algèbre commutative, Lecture notes in math., vol. 32, (1967), Springer-Verlag Berlin · Zbl 0154.01402
[2] André, M., Homologie des algèbres commutatives, Grundlehren math. wiss., Band 206, (1974), Springer-Verlag Berlin · Zbl 0284.18009
[3] Boij, M., Components of the space parametrizing graded Gorenstein Artin algebras with a given Hilbert function, Pacific J. math., 187, 1, 1-11, (1999) · Zbl 0940.13009
[4] Boij, M., Gorenstein Artin algebras and points in projective space, Bull. London math. soc., 31, 1, 11-16, (1999) · Zbl 0940.13008
[5] M. Boij, A. Iarrobino, Families of level algebras, in: Talk by Iarrobino at the Magic ’05 Conference, Notre Dame University, October 2005, in press · Zbl 1165.13009
[6] Bolondi, G.; Kleppe, J.O.; Miro-Roig, R.M., Maximal rank curves and singular points of the Hilbert scheme, Compos. math., 77, 269-291, (1991) · Zbl 0724.14018
[7] R. Buchweitz, Contributions a la theorie des singularites, thesis l’Université Paris VII, 1981
[8] R. Buchweitz, B. Ulrich, Homological properties which are invariant under linkage, preprint, 1983
[9] Chipalkatti, J.V.; Geramita, A.V., On parameter spaces for Artin level algebras, Michigan math. J., 51, 187-207, (2003) · Zbl 1097.13514
[10] Diesel, S.J., Some irreducibility and dimension theorems for families of height 3 Gorenstein algebras, Pacific J. math., 172, 365-397, (1996) · Zbl 0882.13021
[11] Eisenbud, D., Commutative algebra. with a view toward algebraic geometry, Grad. texts in math., vol. 150, (1995), Springer-Verlag New York · Zbl 0819.13001
[12] Ellingsrud, G., Sur le schéma de Hilbert des variétés de codimension 2 dans \(\mathbb{P}^e\) a cône de cohen – macaulay, Ann. sci. école norm. sup., 8, 423-432, (1975) · Zbl 0325.14002
[13] Geramita, A.V.; Maroscia, P.; Roberts, L.G., The Hilbert function of a reduced k-algebra, J. London math. soc., 28, 443-452, (1993) · Zbl 0535.13012
[14] A.V. Geramita, T. Harima, J. Migliore, Y.S. Shin, The Hilbert function of a level algebra, Mem. Amer. Math. Soc., in press · Zbl 1121.13019
[15] Gotzmann, G., A stratification of the Hilbert scheme of points in the projective plane, Math. Z., 199, 4, 539-547, (1988) · Zbl 0637.14003
[16] Grayson, D.; Stillman, M., Macaulay 2—A software system for algebraic geometry and commutative algebra, available at
[17] Grothendieck, A., Techniques de construction et théorèmes d’existence en géométrie algébrique IV: LES schémas des Hilbert, Séminaire bourbaki, 13, 221, (1960/1961) · Zbl 0236.14003
[18] Grothendieck, A., Eléments de la géométrie algébriques IV. etude locale des schémas et des morphismes de schémas, Publ. math. inst. hautes études sci., 32, (1967) · Zbl 0153.22301
[19] Hartshorne, R., Local cohomology, Lecture notes in math., vol. 41, (1967), Springer-Verlag New York
[20] Hartshorne, R., Connectedness of the Hilbert scheme, Publ. math. inst. hautes études sci., 29, 5-48, (1966) · Zbl 0171.41502
[21] Herzog, J., Deformationen von cohen – macaulay algebren, J. reine angew. math., 318, 83-105, (1980) · Zbl 0425.13005
[22] Huneke, C., Numerical invariants of liaison classes, Invent. math., 75, 301-325, (1984) · Zbl 0536.13005
[23] Iarrobino, A., Punctual Hilbert schemes, Mem. amer. math. soc., 10, 188, (1977) · Zbl 0355.14001
[24] Iarrobino, A., Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. amer. math. soc., 285, 1, 337-378, (1984) · Zbl 0548.13009
[25] Iarrobino, A., Hilbert functions of Gorenstein algebras associated to a pencil of forms, () · Zbl 1101.13024
[26] Iarrobino, A.; Kanev, V., Power sums, Gorenstein algebras and determinantal loci, Lecture notes in math., vol. 1721, (1999), Springer-Verlag New York · Zbl 0942.14026
[27] Iarrobino, A.; Srinivasan, H., Some Gorenstein Artin algebras of embedding dimension four, I: components of \(\operatorname{PGor}(H)\) for \(H = (1, 4, 7, \ldots, 1)\), J. pure appl. algebra, 201, 1-3, 62-96, (2005) · Zbl 1107.13020
[28] Kleppe, J.O., Deformations of graded algebras, Math. scand., 45, 205-231, (1979) · Zbl 0436.14004
[29] Kleppe, J.O., Liaison of families of subschemes in \(\mathbb{P}^n\), () · Zbl 0697.14003
[30] Kleppe, J.O., The smoothness and the dimension of \(\operatorname{PGor}(H)\) and of other strata of the punctual Hilbert scheme, J. algebra, 200, 2, 606-628, (1998) · Zbl 0928.14005
[31] Kleppe, J.O.; Miro-Roig, R., The dimension of the Hilbert scheme of Gorenstein codimension 3 subschemes, J. pure appl. algebra, 127, 1, 73-82, (1998) · Zbl 0949.14003
[32] Kleppe, J.O., Maximal families of Gorenstein algebras, Trans. amer. math. soc., 358, 7, 3133-3167, (2006) · Zbl 1103.14005
[33] Kleppe, J.O.; Miro-Roig, R., Dimension of families of determinantal schemes, Trans. amer. math. soc., 357, 7, 2871-2907, (2005) · Zbl 1073.14063
[34] J.O. Kleppe, Unobstructedness and dimension of families of Gorenstein algebras, preprint, August 2004, Collect. Math., in press
[35] Kleppe, J.O., The Hilbert scheme of space curves of small diameter, Ann. inst. Fourier, 56, 5, 1297-1335, (2006) · Zbl 1117.14006
[36] Kleppe, J.O.; Migliore, J.; Miro-Roig, R.; Nagel, U.; Peterson, C., Gorenstein liaison, complete intersection liaison invariants and unobstructedness, Mem. amer. math. soc., 154, 732, (2001) · Zbl 1006.14018
[37] J. Kleppe, Additive splittings of homogeneous polynomials, thesis, June 2005, Univ. of Oslo
[38] Laudal, A., Formal moduli of algebraic structures, Lecture notes in math., vol. 754, (1979), Springer-Verlag New York · Zbl 0438.14007
[39] Lichtenbaum, S.; Schlessinger, M., The cotangent complex of a morphism, Trans. amer. math. soc., 128, 41-70, (1967) · Zbl 0156.27201
[40] Mall, D., Connectedness of the Hilbert function strata and other connectedness results, J. pure appl. algebra, 150, 175-205, (2000) · Zbl 0986.14002
[41] Martin-Deschamps, M.; Perrin, D., Sur la classification des courbes gauches, I, Astérisque, 184-185, (1990)
[42] Martin-Deschamps, M.; Piene, R., Arithmetically cohen – macaulay curves in \(\mathbb{P}^4\) of degree 4 and genus 0, Manuscripta math., 93, 391-408, (1997) · Zbl 0905.14027
[43] Migliore, J., Families of reduced zero-dimensional schemes, Collect. math., 57, 2, 173-192, (2006) · Zbl 1101.13017
[44] Migliore, J., Introduction to liaison theory and deficiency modules, Progr. math., vol. 165, (1998), Birkhäuser Boston Boston, MA · Zbl 0921.14033
[45] Mumford, D., Lectures on curves on an algebraic surface, Ann. of math. stud., vol. 59, (1966), Princeton Univ. Press · Zbl 0187.42701
[46] Notari, R.; Spreafico, M.L., A stratification of Hilbert scheme by initial ideals and applications, Manuscripta math., 101, 429-448, (2000) · Zbl 0985.13006
[47] Pardue, K., Deformation classes of graded modules and maximal Betti numbers, Illinois J. math., 40, 4, 564-585, (1996) · Zbl 0903.13004
[48] Piene, R.; Schlessinger, M., On the Hilbert scheme compactification of the space of twisted cubics. I, Amer. J. math., 107, 761-774, (1985) · Zbl 0589.14009
[49] Ragusa, A.; Zappala, G., On the reducibility of the postulation Hilbert scheme, Rend. circ. mat. Palermo (2), 53, 401-406, (2004) · Zbl 1099.13029
[50] Walter, C., Some examples of obstructed curves in \(\mathbb{P}^3\), () · Zbl 0787.14021
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