The Hilbert scheme of space curves of small diameter

Kleppe, Jan Oddvar

Annales de l'Institut Fourier, Tome 56 (2006) no. 5, pp. 1297-1335.

Cet article concerne des courbes gauches $C$ de degré $d$ et de genre $g$, d’idéal homogène $I$ et de module de Rao $M={\mathrm{H}}_{*}^1\left(\tilde{I}\right)$ ; les résultats principaux portent sur les courbes qui vérifient ${}_0{\mathrm{Ext}}_R^2(M,M)=0$ (e.g. de diamètre, $\mathrm{diam}M\le 2$). Pour de telles courbes nous trouvons des conditions nécessaires et suffisantes pour être non obstruées et nous calculons la dimension du schéma de Hilbert, $\mathrm{H}(d,g)$ en $\left(C\right)$ sous des conditions suffisantes. Dans le cas du diamètre 1, les conditions nécessaires et suffisantes coïncident et la condition d’être non obstruée s’avère être équivalente à l’annulation de certains nombres de Betti gradués de la résolution libre minimale de $I$. Plus généralement, en prenant des déformations convenables de $C$, nous montrons comment éliminer les facteurs directs libres répétés (“termes-fantômes”) dans la résolution minimale de $I$, conduisant à une description relativement concrète du nombre des composantes irréductibles de $\mathrm{H}(d,g)$ qui contiennent une courbe obstruée de diamètre 1. Nous prouvons aussi que chaque composante irréductible de $\mathrm{H}(d,g)$ est réduite au cas de diamètre 1.

This paper studies space curves $C$ of degree $d$ and arithmetic genus $g$, with homogeneous ideal $I$ and Rao module $M={\mathrm{H}}_{*}^1\left(\tilde{I}\right)$, whose main results deal with curves which satisfy ${}_0{\mathrm{Ext}}_R^2(M,M)=0$ (e.g. of diameter, $\mathrm{diam}M\le 2$). For such curves we find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, $\mathrm{H}(d,g)$, at $\left(C\right)$ under the sufficient conditions. In the diameter one case, the necessary and sufficient conditions coincide, and the unobstructedness of $C$ turns out to be equivalent to the vanishing of certain graded Betti numbers of the free minimal resolution of $I$. More generally by taking suitable deformations of $C$ we show how to kill repeated direct free factors (“ghost-terms”) in the minimal resolution of $I$, leading to a rather concrete description of the number of irreducible components of $\mathrm{H}(d,g)$ which contains an obstructed diameter one curve. We also show that every irreducible component of $\mathrm{H}(d,g)$ is reduced in the diameter one case.

Classification : 14C05, 14H50, 14B10, 14B15, 13D10, 13D02, 13D07, 13C40
Keywords: Hilbert scheme, space curve, Buchsbaum curve, unobstructedness, cup-product, graded Betti numbers, ghost term, linkage, normal module, postulation Hilbert scheme
Mot clés : schéma de Hilbert, courbe de l’espace, courbe de Buschsbaum, non obstruction, nombres de Betti gradués, termes-fantômes, lien, module normal, schéma de Hilbert

[1] Amasaki, Mutsumi Examples of nonsingular irreducible curves which give reducible singular points of $\mathrm{red}\left(H_{d,g}\right)$, Publ. RIMS, Kyoto Univ., Volume 21 (1985) no. 4, pp. 761-786 | DOI | MR | Zbl

[2] Bolondi, G.; Migliore, J. C. On curves with natural cohomology and their deficiency modules, Ann. Inst. Fourier (Grenoble), Volume 43 (1993) no. 2, pp. 325-357 | DOI | Numdam | MR | Zbl

[3] Bolondi, Giorgio Irreducible families of curves with fixed cohomology, Arch. Math. (Basel), Volume 53 (1989) no. 3, pp. 300-305 | MR | Zbl

[4] Bolondi, Giorgio; Kleppe, Jan O.; Miró-Roig, Rosa María Maximal rank curves and singular points of the Hilbert scheme, Compositio Math., Volume 77 (1991) no. 3, pp. 269-291 | Numdam | MR | Zbl

[5] Chang, Mei-Chu A filtered Bertini-type theorem, J. Reine Angew. Math., Volume 397 (1989), pp. 214-219 | MR | Zbl

[6] Eisenbud, David Commutative algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995 (With a view toward algebraic geometry) | MR | Zbl

[7] Ellia, Philippe D’autres composantes non réduites de $\mathrm{Hilb}{\mathbf{P}}^3$, Math. Ann., Volume 277 (1987) no. 3, pp. 433-446 | DOI | MR | Zbl

[8] Ellia, Philippe; Fiorentini, Mario Défaut de postulation et singularités du schéma de Hilbert, Ann. Univ. Ferrara Sez. VII (N.S.), Volume 30 (1984), pp. 185-198 | MR | Zbl

[9] Ellia, Philippe; Fiorentini, Mario Quelques remarques sur les courbes arithmétiquement Buchsbaum de l’espace projectif, Ann. Univ. Ferrara Sez. VII (N.S.), Volume 33 (1987), pp. 89-111 | MR | Zbl

[10] Ellingsrud, Geir Sur le schéma de Hilbert des variétés de codimension $2$ dans ${\mathbf{P}}^e$ à cône de Cohen-Macaulay, Ann. Sci. École Norm. Sup. (4), Volume 8 (1975) no. 4, pp. 423-431 | Numdam | MR | Zbl

[11] Fløystad, Gunnar Determining obstructions for space curves, with applications to nonreduced components of the Hilbert scheme, J. Reine Angew. Math., Volume 439 (1993), pp. 11-44 | DOI | MR | Zbl

[12] Ginouillac, Stéphane Sur le nombre de composantes du schéma de Hilbert des courbes ACM de ${\mathbf{P}}_{\mathbf{k}}^3$, C. R. Acad. Sci. Paris Sér. I Math., Volume 329 (1999) no. 10, pp. 857-862 | DOI | MR | Zbl

[13] Grothendieck, A. Les schémas de Hilbert, Séminaire Bourbaki, Volume 221, Secrétariat mathématique, 1960 | Numdam

[14] Grothendieck, Alexander Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux $(SGA$ $2)$, North-Holland Publishing Co., Amsterdam, 1968 (Augmenté d’un exposé par Michèle Raynaud, Séminaire de Géométrie Algébrique du Bois-Marie, 1962, Advanced Studies in Pure Mathematics, Vol. 2) | MR | Zbl

[15] Gruson, Laurent; Peskine, Christian Genre des courbes de l’espace projectif. II, Ann. Sci. École Norm. Sup. (4), Volume 15 (1982) no. 3, pp. 401-418 | Numdam | MR | Zbl

[16] Guffroy, Sébastien 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 | Zbl

[17] Hartshorne, Robin Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York, 1983 | MR | Zbl

[18] Kleppe, Jan O. The Hilbert-flag scheme, its properties and its connection with the Hilbert scheme. Applications to curves in 3-space Preprint (part of thesis), March 1981, Univ. of Oslo

[19] Kleppe, Jan O. The Hilbert scheme of space curves of small Rao module with an appendix on non-reduced components (Preprint June 1996) | MR

[20] Kleppe, Jan O. Deformations of graded algebras, Math. Scand., Volume 45 (1979) no. 2, pp. 205-231 | MR | Zbl

[21] Kleppe, Jan O. Nonreduced components of the Hilbert scheme of smooth space curves, Space curves (Rocca di Papa, 1985) (Lecture Notes in Math.), Volume 1266, Springer, Berlin, 1987, pp. 181-207 | MR | Zbl

[22] Kleppe, Jan O. Liaison of families of subschemes in ${\mathbf{P}}^n$, Algebraic curves and projective geometry (Trento, 1988) (Lecture Notes in Math.), Volume 1389, Springer, Berlin, 1989, pp. 128-173 | MR | Zbl

[23] Kleppe, Jan O. Concerning the existence of nice components in the Hilbert scheme of curves in ${\mathbf{P}}^n$ for $n=4$ and $5$, J. Reine Angew. Math., Volume 475 (1996), pp. 77-102 | DOI | MR | Zbl

[24] Laudal, O. A. Matric Massey products and formal moduli. I, Algebra, algebraic topology and their interactions (Stockholm, 1983) (Lecture Notes in Math.), Volume 1183, Springer, Berlin, 1986, pp. 218-240 | MR | Zbl

[25] Laudal, Olav Arnfinn Formal moduli of algebraic structures, Lecture Notes in Mathematics, 754, Springer, Berlin, 1979 | MR | Zbl

[26] Martin-Deschamps, Mireille; Perrin, Daniel Sur la classification des courbes gauches, Astérisque, Volume 184-185 (1990) no. 208 | Numdam | MR | Zbl

[27] Martin-Deschamps, Mireille; Perrin, Daniel Courbes gauches et modules de Rao, J. Reine Angew. Math., Volume 439 (1993), pp. 103-145 | MR | Zbl

[28] Martin-Deschamps, Mireille; Perrin, Daniel Sur les courbes gauches à modules de Rao non connexes, C. R. Acad. Sci. Paris Sér. I Math., Volume 319 (1994) no. 3, pp. 233-236 | MR | Zbl

[29] Martin-Deschamps, Mireille; Perrin, Daniel Le schéma de Hilbert des courbes gauches localement Cohen-Macaulay n’est (presque) jamais réduit, Ann. Sci. École Norm. Sup. (4), Volume 29 (1996) no. 6, pp. 757-785 | Numdam | MR | Zbl

[30] Migliore, Juan C. Introduction to liaison theory and deficiency modules, Progress in Mathematics, 165, Birkhäuser Boston Inc., Boston, MA, 1998 | MR | Zbl

[31] Migliore, Juan C. Families of reduced zero-dimensional schemes, Collect. Math., Volume 57 (2006) no. 2, pp. 173-192 | MR | Zbl

[32] Migliore, Juan C.; Nagel, U. Tetrahedral curves, Int. Math. Res. Not. (2005) no. 15, pp. 899-939 | DOI | MR | Zbl

[33] Miró-Roig, Rosa M. Unobstructed arithmetically Buchsbaum curves, Algebraic curves and projective geometry (Trento, 1988) (Lecture Notes in Math.), Volume 1389, Springer, Berlin, 1989, pp. 235-241 | MR | Zbl

[34] Mumford, David Further pathologies in algebraic geometry, Amer. J. Math., Volume 84 (1962), pp. 642-648 | DOI | MR | Zbl

[35] Nasu, H. Obstructions to deforming space curves and non-reduced components of the Hilbert scheme, Publ. RIMS, Kyoto Univ. (2006) no. 42, pp. 117-141 | DOI | MR | Zbl

[36] Peeva, Irena Consecutive cancellations in Betti numbers, Proc. Amer. Math. Soc., Volume 132 (2004) no. 12, pp. 3503-3507 | DOI | MR | Zbl

[37] Prabhakar Rao, A. Liaison among curves in ${\mathbf{P}}^3$, Invent. Math., Volume 50 (1978/79) no. 3, pp. 205-217 | DOI | MR | Zbl

[38] Sernesi, E. Un esempio di curva ostruita in ${\mathbb{P}}^3$, Sem. di variabili Complesse, Bologna (1981), pp. 223-231 | MR

[39] Walter, Charles H. Horrocks theory and algebraic space curves (Preprint 1990)

[40] Walter, Charles H. Some examples of obstructed curves in ${\mathbb{P}}^3$, Complex projective geometry (Trieste, 1989/Bergen, 1989) (London Math. Soc. Lecture Note Ser.), Volume 179, Cambridge Univ. Press, Cambridge, 1992, pp. 324-340 | MR | Zbl