no. 6
The Hilbert Scheme of Buchsbaum space curves
[Le schéma de Hilbert des courbes gauches de Buchsbaum]
Kleppe, Jan O.1
1 Oslo and Akershus University College of Applied Sciences Faculty of Technology, Art and Design Pb. 4, St. Olavs plass N-0130 Oslo, Norway
Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2099-2130.

Nous considérons le schéma de Hilbert H(d,g) des courbes C dont l’idéal homogène est I(C):=H * 0 (ℐ C ) et le module de Rao M:=H * 1 (ℐ C ). En prenant des générisations (déformations) convenables C ′ de C on simplifie la résolution minimale libre de I(C). Par exemple, certains facteurs libres consécutifs vont disparaître dans une résolution libre de I(C ′ ). En appliquant ceci à des courbes de Buchsbaum de diamètre 1 (M v ≠0 seulement pour une valeur de v), nous donnons une correspondance biunivoque entre l’ensemble 𝒮 des composantes irréductibles de H(d,g) qui contiennent (C) et un ensemble des quintuplets minimaux, qui se spécialise à un quintuple de nombres de Betti gradués de C. De plus nous déterminons presque complétement les nombres de Betti gradués de toutes les générisations de C, et nous donnons une description du lieu singulier du schéma de Hilbert des courbes de diamètre au plus égal à 1. Nous démontrons aussi des résultats de sémi-continuité pour les nombres de Betti gradués des courbes.

We consider the Hilbert scheme H(d,g) of space curves C with homogeneous ideal I(C):=H * 0 (ℐ C ) and Rao module M:=H * 1 (ℐ C ). By taking suitable generizations (deformations to a more general curve) C ′ of C, we simplify the minimal free resolution of I(C) by e.g making consecutive free summands (ghost-terms) disappear in a free resolution of I(C ′ ). Using this for Buchsbaum curves of diameter one (M v ≠0 for only one v), we establish a one-to-one correspondence between the set 𝒮 of irreducible components of H(d,g) that contain (C) and a set of minimal 5-tuples that specializes in an explicit manner to a 5-tuple of certain graded Betti numbers of C related to ghost-terms. Moreover we almost completely (resp. completely) determine the graded Betti numbers of all generizations of C (resp. all generic curves of 𝒮), and we give a specific description of the singular locus of the Hilbert scheme of curves of diameter at most one. We also prove some semi-continuity results for the graded Betti numbers of any space curve under some assumptions.

DOI : 10.5802/aif.2744
Classification : 14C05, 14H50, 14M06, 13D02, 13C40
Keywords: Hilbert scheme, space curve, Buchsbaum curve, graded Betti numbers, ghost term, linkage.
Mot clés : schéma de Hilbert, courbe, courbe de Buchsbaum, nombre de Betti gradué.

Kleppe, Jan O. 1

1 Oslo and Akershus University College of Applied Sciences Faculty of Technology, Art and Design Pb. 4, St. Olavs plass N-0130 Oslo, Norway 
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Kleppe, Jan O. The Hilbert Scheme of Buchsbaum space curves. Annales de l'Institut Fourier, Tome 62 (2012) no. 6, pp. 2099-2130. doi : 10.5802/aif.2744. https://aif.centre-mersenne.org/articles/10.5802/aif.2744/

[1] Amasaki, M. Examples of Nonsingular Irreducible Curves Which Give Reducible Singular Points of red(H d,g ), Publ. RIMS, Kyoto Univ., Volume 21 (1985), pp. 761-786 | MR | Zbl

[2] Bolondi, G. Irreducible families of curves with fixed cohomology, Arch. der Math., Volume 53 (1989), pp. 300-305 | MR | Zbl

[3] Bolondi, G.; Kleppe, J.O.; Miro-Roig, R.M. Maximal rank curves and singular points of the Hilbert scheme, Compositio Math., Volume 77 (1991), pp. 269-291 | Numdam | MR | Zbl

[4] Bolondi, G.; Migliore, J. Classification of Maximal Rank Curves in the Liaison Class L n , Math. Ann., Volume 277 (1987), pp. 585-603 | MR | Zbl

[5] Bolondi, G.; Migliore, J. The Lazarsfeld-Rao property on an arithmetically Gorenstein variety, Manuscripta Math., Volume 78 (1993) no. 4, pp. 347-368 | MR | Zbl

[6] Boratyński, M.; Greco, S. Hilbert functions and Betti numbers in a flat family, Ann. Mat. Pura Appl. (4), Volume 142 (1985), pp. 277-292 | MR | Zbl

[7] Chang, M-C. A Filtered Bertini-type Theorem, J. reine angew. Math., Volume 397 (1989), pp. 214-219 | MR | Zbl

[8] Eisenbud, D. Commutative algebra. With a view toward algebraic geometry, Graduate Texts in Math., 150, Springer-Verlag, New York, 1995 | MR | Zbl

[9] Ellia, Ph.; Fiorentini, M. Défaut de postulation et singularités du Schéma de Hilbert, Annali Univ. di Ferrara, Volume 30 (1984), pp. 185-198 | MR | Zbl

[10] Ellia, Ph.; Hirschowitz, A.; Mezzetti, E. On the number of irreducible components of the Hilbert scheme of smooth space curves, International J. of Math., Volume 3 (1992) no. 6, pp. 799-807 | MR | Zbl

[11] Ellingsrud, G. Sur le schéma de Hilbert des variétés de codimension 2 dans ℙ e à cône de Cohen-Macaulay, Ann. Scient. Éc. Norm. Sup., Volume 8 (1975), pp. 423-432 | Numdam | MR | Zbl

[12] Ginouillac, S. Sur le nombre de composant du schéma de Hilbert des courbes ACM de ℙ k 3 , C. R. Acad. Sci. Paris Sér. I, Math., Volume 329 (1999), pp. 857-862 | MR | Zbl

[13] Grothendieck, A. Les schémas de Hilbert, Séminaire Bourbaki, exp. 221, 1960

[14] Gruson, L.; Peskine, Chr. Genre des courbes de l’espace projectif, Proc. Tromsø 1977 (Lectures Notes in Math.), Volume 687, Springer-Verlag, New York, 1978 | MR | Zbl

[15] Guffroy, S. Sur l’incomplétude de la série linéaire caractéristique d’une famille de courbes planes à nœuds et à cusps, Nagoya Math. J., Volume 171 (2003), pp. 51-83 | MR | Zbl

[16] Hartshorne, R. Algebraic Geometry, Graduate Texts in Math., 52, Springer-Verlag, New York, 1983 | MR | Zbl

[17] Iarrobino, A. Betti strata of height two ideals, J. Algebra, Volume 285 (2005), pp. 835-855 | MR | Zbl

[18] Kleppe, J. O. Deformations of graded algebras, Math. Scand., Volume 45 (1979), pp. 205-231 | MR | Zbl

[19] Kleppe, J. O. Liaison of families of subschemes in ℙ n , Algebraic Curves and Projective Geometry, Proceedings (Trento, 1988) (Lectures Notes in Math.), Volume 1389, Springer-Verlag, New York, 1989 | MR | Zbl

[20] Kleppe, J. O. The Hilbert Scheme of Space Curves of small diameter, Annales de l’institut Fourier, Volume 56 (2006) no. 5, pp. 1297-1335 | Numdam | MR | Zbl

[21] Kleppe, J. O. Families of Artinian and one-dimensional algebras, J. Algebra, Volume 311 (2007), pp. 665-701 | MR | Zbl

[22] Laudal, A. Formal Moduli of Algebraic Structures, Lectures Notes in Math., 754, Springer-Verlag, New York, 1979 | MR | Zbl

[23] Martin-Deschamps, M.; Perrin, D. Sur la classification des courbes gauches, Asterisque, 184-185, 1990 | MR | Zbl

[24] Martin-Deschamps, M.; Perrin, D. Courbes Gauches et Modules de Rao, J. reine angew. Math., Volume 439 (1993), pp. 103-145 | MR | Zbl

[25] Migliore, J. Introduction to liaison theory and deficiency modules, Progress in Math., 165, Birkhäuser Boston, Inc., Boston, MA, 1998 | MR | Zbl

[26] Peeva, I. Consecutive cancellations in Betti Numbers, Proc. Amer. Math. Soc., Volume 132 (2004) no. 12, pp. 3503-3507 | MR | Zbl

[27] Peskine, Ch.; Szpiro, L. Liaison des variétés algébrique, Invent. Math., Volume 26 (1974), pp. 271-302 | MR | Zbl

[28] Piene, R.; Schlessinger, M. On the Hilbert scheme compactification of the space of twisted cubics, Amer. J. Math., Volume 107 (1985), pp. 761-774 | MR | Zbl

[29] Ragusa, A.; Zappalá, G. On the reducibility of the postulation Hilbert scheme, Rend. Circ. Mat. Palermo, Serie II (2004) no. LIII, pp. 401-406 | MR | Zbl

[30] Rao, A. P. Liaison Among Curves, Invent. Math., Volume 50 (1979), pp. 205-217 | MR | Zbl

[31] Sernesi, E. Un esempio di curva ostruita in ℙ 3 (1981), pp. 223-231

[32] Walter, C. Some examples of obstructed curves in ℙ 3 , Complex Projective Geometry (London Math. Soc. Lecture Note Ser.), Volume 179, 1992 | MR | Zbl

[33] Walter, C. Transversality Theorems in General Characteristic with Applications to Arithmetically Buchsbaum Schemes, Internat. J. Math., Volume 5 (1994), pp. 609-617 | MR | Zbl

[34] Weibel, C. A. An introduction to homological algebra, Cambridge Studies in Advanced Mathematics, 38, Cambridge University Press, 1994 | MR | Zbl

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