The Hilbert Scheme of Buchsbaum space curves

Kleppe, Jan O.

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 $I\left(C\right):=H_{*}^0\left({ℐ}_C\right)$ et le module de Rao $M:=H_{*}^1\left({ℐ}_C\right)$. En prenant des générisations (déformations) convenables $C^{\prime }$ de $C$ on simplifie la résolution minimale libre de $I\left(C\right)$. Par exemple, certains facteurs libres consécutifs vont disparaître dans une résolution libre de $I\left(C^{\prime }\right)$. En appliquant ceci à des courbes de Buchsbaum de diamètre 1 ($M_v\ne 0$ seulement pour une valeur de $v$), nous donnons une correspondance biunivoque entre l’ensemble $𝒮$ des composantes irréductibles de $H(d,g)$ qui contiennent $\left(C\right)$ 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 $I\left(C\right):=H_{*}^0\left({ℐ}_C\right)$ and Rao module $M:=H_{*}^1\left({ℐ}_C\right)$. By taking suitable generizations (deformations to a more general curve) $C^{\prime }$ of $C$, we simplify the minimal free resolution of $I\left(C\right)$ by e.g making consecutive free summands (ghost-terms) disappear in a free resolution of $I\left(C^{\prime }\right)$. Using this for Buchsbaum curves of diameter one ($M_v\ne 0$ for only one $v$), we establish a one-to-one correspondence between the set $𝒮$ of irreducible components of $H(d,g)$ that contain $\left(C\right)$ 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.

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.

[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 ${\mathbb{P}}^e$ 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 ${\mathbb{P}}_k^3$, 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 ${\mathbb{P}}^n$, 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 | Numdam | 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 ${\mathbb{P}}^3$ (1981), pp. 223-231

[32] Walter, C. Some examples of obstructed curves in ${\mathbb{P}}^3$, 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