This paper studies space curves of degree and arithmetic genus , with homogeneous ideal and Rao module , whose main results deal with curves which satisfy (e.g. of diameter, ). For such curves we find necessary and sufficient conditions for unobstructedness, and we compute the dimension of the Hilbert scheme, , at under the sufficient conditions. In the diameter one case, the necessary and sufficient conditions coincide, and the unobstructedness of turns out to be equivalent to the vanishing of certain graded Betti numbers of the free minimal resolution of . More generally by taking suitable deformations of we show how to kill repeated direct free factors (“ghost-terms”) in the minimal resolution of , leading to a rather concrete description of the number of irreducible components of which contains an obstructed diameter one curve. We also show that every irreducible component of is reduced in the diameter one case.
Cet article concerne des courbes gauches de degré et de genre , d’idéal homogène et de module de Rao ; les résultats principaux portent sur les courbes qui vérifient (e.g. de diamètre, ). Pour de telles courbes nous trouvons des conditions nécessaires et suffisantes pour être non obstruées et nous calculons la dimension du schéma de Hilbert, en 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 . Plus généralement, en prenant des déformations convenables de , nous montrons comment éliminer les facteurs directs libres répétés (“termes-fantômes”) dans la résolution minimale de , conduisant à une description relativement concrète du nombre des composantes irréductibles de qui contiennent une courbe obstruée de diamètre 1. Nous prouvons aussi que chaque composante irréductible de est réduite au cas de diamètre 1.
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
@article{AIF_2006__56_5_1297_0, author = {Kleppe, Jan Oddvar}, title = {The {Hilbert} scheme of space curves of small diameter}, journal = {Annales de l'Institut Fourier}, pages = {1297--1335}, publisher = {Association des Annales de l{\textquoteright}institut Fourier}, volume = {56}, number = {5}, year = {2006}, doi = {10.5802/aif.2214}, zbl = {1117.14006}, mrnumber = {2273858}, language = {en}, url = {http://archive.numdam.org/articles/10.5802/aif.2214/} }
TY - JOUR AU - Kleppe, Jan Oddvar TI - The Hilbert scheme of space curves of small diameter JO - Annales de l'Institut Fourier PY - 2006 SP - 1297 EP - 1335 VL - 56 IS - 5 PB - Association des Annales de l’institut Fourier UR - http://archive.numdam.org/articles/10.5802/aif.2214/ DO - 10.5802/aif.2214 LA - en ID - AIF_2006__56_5_1297_0 ER -
%0 Journal Article %A Kleppe, Jan Oddvar %T The Hilbert scheme of space curves of small diameter %J Annales de l'Institut Fourier %D 2006 %P 1297-1335 %V 56 %N 5 %I Association des Annales de l’institut Fourier %U http://archive.numdam.org/articles/10.5802/aif.2214/ %R 10.5802/aif.2214 %G en %F AIF_2006__56_5_1297_0
Kleppe, Jan Oddvar. The Hilbert scheme of space curves of small diameter. Annales de l'Institut Fourier, Volume 56 (2006) no. 5, pp. 1297-1335. doi : 10.5802/aif.2214. http://archive.numdam.org/articles/10.5802/aif.2214/
[1] Examples of nonsingular irreducible curves which give reducible singular points of , Publ. RIMS, Kyoto Univ., Volume 21 (1985) no. 4, pp. 761-786 | DOI | MR | Zbl
[2] 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] Irreducible families of curves with fixed cohomology, Arch. Math. (Basel), Volume 53 (1989) no. 3, pp. 300-305 | MR | Zbl
[4] Maximal rank curves and singular points of the Hilbert scheme, Compositio Math., Volume 77 (1991) no. 3, pp. 269-291 | Numdam | MR | Zbl
[5] A filtered Bertini-type theorem, J. Reine Angew. Math., Volume 397 (1989), pp. 214-219 | MR | Zbl
[6] Commutative algebra, Graduate Texts in Mathematics, 150, Springer-Verlag, New York, 1995 (With a view toward algebraic geometry) | MR | Zbl
[7] D’autres composantes non réduites de , Math. Ann., Volume 277 (1987) no. 3, pp. 433-446 | DOI | MR | Zbl
[8] 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] 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] Sur le schéma de Hilbert des variétés de codimension dans à cône de Cohen-Macaulay, Ann. Sci. École Norm. Sup. (4), Volume 8 (1975) no. 4, pp. 423-431 | Numdam | MR | Zbl
[11] 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] Sur le nombre de composantes du schéma de Hilbert des courbes ACM de , C. R. Acad. Sci. Paris Sér. I Math., Volume 329 (1999) no. 10, pp. 857-862 | DOI | MR | Zbl
[13] Les schémas de Hilbert, Séminaire Bourbaki, Volume 221, Secrétariat mathématique, 1960 | Numdam
[14] Cohomologie locale des faisceaux cohérents et théorèmes de Lefschetz locaux et globaux , 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] 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] 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] Algebraic Geometry, Graduate Texts in Mathematics, 52, Springer-Verlag, New York, 1983 | MR | Zbl
[18] 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] The Hilbert scheme of space curves of small Rao module with an appendix on non-reduced components (Preprint June 1996) | MR
[20] Deformations of graded algebras, Math. Scand., Volume 45 (1979) no. 2, pp. 205-231 | MR | Zbl
[21] 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] Liaison of families of subschemes in , Algebraic curves and projective geometry (Trento, 1988) (Lecture Notes in Math.), Volume 1389, Springer, Berlin, 1989, pp. 128-173 | MR | Zbl
[23] Concerning the existence of nice components in the Hilbert scheme of curves in for and , J. Reine Angew. Math., Volume 475 (1996), pp. 77-102 | DOI | MR | Zbl
[24] 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] Formal moduli of algebraic structures, Lecture Notes in Mathematics, 754, Springer, Berlin, 1979 | MR | Zbl
[26] Sur la classification des courbes gauches, Astérisque, Volume 184-185 (1990) no. 208 | Numdam | MR | Zbl
[27] Courbes gauches et modules de Rao, J. Reine Angew. Math., Volume 439 (1993), pp. 103-145 | MR | Zbl
[28] 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] 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] Introduction to liaison theory and deficiency modules, Progress in Mathematics, 165, Birkhäuser Boston Inc., Boston, MA, 1998 | MR | Zbl
[31] Families of reduced zero-dimensional schemes, Collect. Math., Volume 57 (2006) no. 2, pp. 173-192 | MR | Zbl
[32] Tetrahedral curves, Int. Math. Res. Not. (2005) no. 15, pp. 899-939 | DOI | MR | Zbl
[33] 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] Further pathologies in algebraic geometry, Amer. J. Math., Volume 84 (1962), pp. 642-648 | DOI | MR | Zbl
[35] 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] Consecutive cancellations in Betti numbers, Proc. Amer. Math. Soc., Volume 132 (2004) no. 12, pp. 3503-3507 | DOI | MR | Zbl
[37] Liaison among curves in , Invent. Math., Volume 50 (1978/79) no. 3, pp. 205-217 | DOI | MR | Zbl
[38] Un esempio di curva ostruita in , Sem. di variabili Complesse, Bologna (1981), pp. 223-231 | MR
[39] Horrocks theory and algebraic space curves (Preprint 1990)
[40] Some examples of obstructed curves in , 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
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