# 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.

##### MSC:
 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)
##### Keywords:
Hilbert function; Betti numbers; socle; Grassmannian variety
Full Text:
##### References:
  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  M. Boij, Betti number strata of the space of codimension three Gorenstein Artin algebras, preprint, 2001  Bruns, U.; Vetter, W., Determinantal rings, Lecture notes in math., vol. 1327, (1988), Springer Berlin · Zbl 0673.13006  Burch, L., On ideals of finite homological dimension in local rings, Proc. Cambridge philos. soc., 64, 941-948, (1968) · Zbl 0172.32302  Campanella, G., Standard bases of perfect homogeneous polynomial ideals of height two, J. algebra, 101, 47-60, (1986) · Zbl 0609.13001  Chipalkatti, J.; Geramita, A., On parameter spaces for Artin level algebras, Michigan math. J., 51, 187-207, (2003) · Zbl 1097.13514  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  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  Emsalem, J.; Iarrobino, A., Inverse system of a symbolic power I, J. algebra, 174, 1080-1090, (1995) · Zbl 0842.14002  Fröberg, R., An inequality for Hilbert series of graded algebras, Math. scand., 56, 117-144, (1985) · Zbl 0582.13007  Geramita, A.; Migliore, J., Hyperplane sections of a smooth curve in $$\mathbb{P}^3$$, Comm. algebra, 17, 3129-3164, (1989) · Zbl 0705.14030  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  Gotzmann, G., A stratification of the Hilbert scheme of points in the projective plane, Math Z., 199, 4, 539-547, (1988) · Zbl 0637.14003  Gruson, L.; Peskine, C., Genre des courbes de l’espace projectif, (), 31-59 · Zbl 0517.14007  Harima, T., Some examples of unimodal Gorenstein sequences, J. pure appl. algebra, 103, 313-324, (1995) · Zbl 0847.13003  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  Iarrobino, A., Punctual Hilbert schemes, Mem. amer. math. soc., 10, 188, (1977) · Zbl 0355.14001  Iarrobino, A., Compressed algebras: Artin algebras having given socle degrees and maximal length, Trans. amer. math. soc., 285, 337-378, (1984) · Zbl 0548.13009  Iarrobino, A., Ancestor ideals of vector spaces of forms, and level algebras, J. algebra, 272, 530-580, (2004) · Zbl 1119.13015  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  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  Kleppe, J.O., Deformations of graded algebras, Math. scand., 45, 205-231, (1979) · Zbl 0436.14004  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  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  Sauer, T., Smoothing projectively cohen – macaulay space curves, Math. ann., 272, 83-90, (1985) · Zbl 0546.14023  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.