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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
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##### References:
 [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
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