zbMATH — the first resource for mathematics

Betti strata of height two ideals. (English) Zbl 1095.13014
Let \(R\) be the polynomial ring in two variables over an infinite field \(k\), and \(H\) the Hilbert function of some graded quotients of \(R\). The variety \(G(H)\) parametrizes graded ideals \(I\) in \(R\) of Hilbert function \(H(R/I,-)=H\). It is a closed subvariety of a finite product of Grassmannians. The Betti stratum \(G_{\beta}(H)\) parametrizes all graded quotients \(R/I\) of \(R\) having fixed \(\beta_i=\dim_k\text{Tor}_1^R(I,k)_i\). The Betti strata are irreducible and Cod\((G_{\beta}(H))=\Sigma_{i>\mu}\beta_i\nu_i\), where \(\mu=\min\{i:H_i<i+1\}\) and \(\nu_i\) is the number of generators of degree \(i\) in \(I\) (\(H\) and \(\beta\) determine uniquely \(\nu\)). When \(k\) is algebraically closed the closure of a Betti stratum is the union of more special strata and is Cohen-Macaulay. Given the socle type, all possible Hilbert functions of Artinian graded quotients of \(R\) are determined.

13D40 Hilbert-Samuel and Hilbert-Kunz functions; Poincaré series
14M15 Grassmannians, Schubert varieties, flag manifolds
13H10 Special types (Cohen-Macaulay, Gorenstein, Buchsbaum, etc.)
13D02 Syzygies, resolutions, complexes and commutative rings
14C05 Parametrization (Chow and Hilbert schemes)
Full Text: DOI arXiv
[1] Arbarello, E.; Cornalba, M.; Griffiths, P.A.; Harris, J., Geometry of algebraic curves, vol. 1, Grundlehren math. wiss., vol. 267, (1985), Springer-Verlag New York · Zbl 0559.14017
[2] M. Boij, Betti number strata of the space of codimension three Gorenstein Artin algebras, preprint, 2001
[3] Bruns, U.; Vetter, W., Determinantal rings, Lecture notes in math., vol. 1327, (1988), Springer Berlin · Zbl 0673.13006
[4] Burch, L., On ideals of finite homological dimension in local rings, Proc. Cambridge philos. soc., 64, 941-948, (1968) · Zbl 0172.32302
[5] Campanella, G., Standard bases of perfect homogeneous polynomial ideals of height two, J. algebra, 101, 47-60, (1986) · Zbl 0609.13001
[6] Chipalkatti, J.; Geramita, A., On parameter spaces for Artin level algebras, Michigan math. J., 51, 187-207, (2003) · Zbl 1097.13514
[7] 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
[8] Ellingsrud, G., Sur le schéma de Hilbert des variétés de codimension 2 dans \(\mathbb{P}^e\) a cône de cohen – macaulay, Ann. sci. ecole norm. sup. (4), 8, 423-432, (1975) · Zbl 0325.14002
[9] Emsalem, J.; Iarrobino, A., Inverse system of a symbolic power I, J. algebra, 174, 1080-1090, (1995) · Zbl 0842.14002
[10] Fröberg, R., An inequality for Hilbert series of graded algebras, Math. scand., 56, 117-144, (1985) · Zbl 0582.13007
[11] Geramita, A.; Migliore, J., Hyperplane sections of a smooth curve in \(\mathbb{P}^3\), Comm. algebra, 17, 3129-3164, (1989) · Zbl 0705.14030
[12] Geramita, A.; Migliore, J.; Pucci, M.; Shin, Y.S., Smooth points of \(\operatorname{Gor}(T)\), J. pure appl. algebra, 122, 209-241, (1997) · Zbl 0905.13004
[13] Gotzmann, G., A stratification of the Hilbert scheme of points in the projective plane, Math Z., 199, 4, 539-547, (1988) · Zbl 0637.14003
[14] Gruson, L.; Peskine, C., Genre des courbes de l’espace projectif, (), 31-59 · Zbl 0517.14007
[15] Harima, T., Some examples of unimodal Gorenstein sequences, J. pure appl. algebra, 103, 313-324, (1995) · Zbl 0847.13003
[16] Herzog, J.; Trung, N.V.; Valla, G., On hyperplane sections of reduced irreducible varieties of low codimension, J. math. Kyoto univ., 34, 1, 47-72, (1994) · Zbl 0836.14031
[17] Iarrobino, A., Punctual Hilbert schemes, Mem. amer. math. soc., 10, 188, (1977) · Zbl 0355.14001
[18] Iarrobino, A., Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. amer. math. soc., 285, 337-378, (1984) · Zbl 0548.13009
[19] Iarrobino, A., Ancestor ideals of vector spaces of forms, and level algebras, J. algebra, 272, 530-580, (2004) · Zbl 1119.13015
[20] Iarrobino, A.; Kanev, V., Power sums, Gorenstein algebras, and determinantal loci, Lecture notes in math., vol. 1721, (1999), Springer Heidelberg, 345+xxvii pp · Zbl 0942.14026
[21] Iarrobino, A.; Kanev, V.; Yaméogo, J., The family \(G_T\) of graded Artinian quotients of \(k [x, y]\) of given Hilbert function, Comm. algebra, 31, 8, 3863-3916, (2003) · Zbl 1048.14003
[22] Kleppe, J.O., Deformations of graded algebras, Math. scand., 45, 205-231, (1979) · Zbl 0436.14004
[23] Maggione, R.; Ragusa, A., The Hilbert function of generic plane sections of curves of \(\mathbb{P}^3\), Invent. math., 91, 2, 253-258, (1988) · Zbl 0616.14026
[24] Piene, R.; Schlessinger, M., On the Hilbert scheme compactification of the space of twisted cubics, Amer. J. math, 107, 761-774, (1985) · Zbl 0589.14009
[25] Sauer, T., Smoothing projectively cohen – macaulay space curves, Math. ann., 272, 83-90, (1985) · Zbl 0546.14023
[26] Weyman, J., Cohomology of vector bundles and syzygies, Cambridge tracts in math., vol. 149, (2003), Cambridge Univ. Press · Zbl 1075.13007
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.