The Hilbert Scheme of Buchsbaum space curves

Jan O. Kleppe[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 (2012)

  • Volume: 62, Issue: 6, page 2099-2130
  • ISSN: 0373-0956

Abstract

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We consider the Hilbert scheme of space curves with homogeneous ideal and Rao module . By taking suitable generizations (deformations to a more general curve) of , we simplify the minimal free resolution of by e.g making consecutive free summands (ghost-terms) disappear in a free resolution of . Using this for Buchsbaum curves of diameter one ( for only one ), we establish a one-to-one correspondence between the set of irreducible components of that contain and a set of minimal 5-tuples that specializes in an explicit manner to a 5-tuple of certain graded Betti numbers of related to ghost-terms. Moreover we almost completely (resp. completely) determine the graded Betti numbers of all generizations of (resp. all generic curves of ), 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.

How to cite

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Kleppe, Jan O.. "The Hilbert Scheme of Buchsbaum space curves." Annales de l’institut Fourier 62.6 (2012): 2099-2130. <http://eudml.org/doc/251108>.

@article{Kleppe2012,
abstract = {We consider the Hilbert scheme $\mathop \{H\}(d,g)$ of space curves $C$ with homogeneous ideal $I(C):=H_\{*\}^0(\mathcal\{I\}_\{C\})$ and Rao module $M:=H_\{*\}^1(\mathcal\{I\}_\{C\})$. By taking suitable generizations (deformations to a more general curve) $C^\{\prime\}$ 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 $I(C^\{\prime\})$. 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 $\mathcal\{S\}$ of irreducible components of $\mathop \{H\}(d,g)$ 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 $\mathcal\{S\}$), 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.},
affiliation = {Oslo and Akershus University College of Applied Sciences Faculty of Technology, Art and Design Pb. 4, St. Olavs plass N-0130 Oslo, Norway},
author = {Kleppe, Jan O.},
journal = {Annales de l’institut Fourier},
keywords = {Hilbert scheme; space curve; Buchsbaum curve; graded Betti numbers; ghost term; linkage},
language = {eng},
number = {6},
pages = {2099-2130},
publisher = {Association des Annales de l’institut Fourier},
title = {The Hilbert Scheme of Buchsbaum space curves},
url = {http://eudml.org/doc/251108},
volume = {62},
year = {2012},
}

TY - JOUR
AU - Kleppe, Jan O.
TI - The Hilbert Scheme of Buchsbaum space curves
JO - Annales de l’institut Fourier
PY - 2012
PB - Association des Annales de l’institut Fourier
VL - 62
IS - 6
SP - 2099
EP - 2130
AB - We consider the Hilbert scheme $\mathop {H}(d,g)$ of space curves $C$ with homogeneous ideal $I(C):=H_{*}^0(\mathcal{I}_{C})$ and Rao module $M:=H_{*}^1(\mathcal{I}_{C})$. By taking suitable generizations (deformations to a more general curve) $C^{\prime}$ 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 $I(C^{\prime})$. 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 $\mathcal{S}$ of irreducible components of $\mathop {H}(d,g)$ 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 $\mathcal{S}$), 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.
LA - eng
KW - Hilbert scheme; space curve; Buchsbaum curve; graded Betti numbers; ghost term; linkage
UR - http://eudml.org/doc/251108
ER -

References

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  1. M. Amasaki, Examples of Nonsingular Irreducible Curves Which Give Reducible Singular Points of red(), Publ. RIMS, Kyoto Univ. 21 (1985), 761-786 Zbl0609.14002MR817163
  2. G. Bolondi, Irreducible families of curves with fixed cohomology, Arch. der Math. 53 (1989), 300-305 Zbl0658.14005MR1006724
  3. G. Bolondi, J.O. Kleppe, R.M. Miro-Roig, Maximal rank curves and singular points of the Hilbert scheme, Compositio Math. 77 (1991), 269-291 Zbl0724.14018MR1092770
  4. G. Bolondi, J. Migliore, Classification of Maximal Rank Curves in the Liaison Class , Math. Ann. 277 (1987), 585-603 Zbl0607.14015MR901706
  5. G. Bolondi, J. Migliore, The Lazarsfeld-Rao property on an arithmetically Gorenstein variety, Manuscripta Math. 78 (1993), 347-368 Zbl0813.14035MR1208646
  6. M. Boratyński, S. Greco, Hilbert functions and Betti numbers in a flat family, Ann. Mat. Pura Appl. (4) 142 (1985), 277-292 Zbl0591.14007MR839041
  7. M-C. Chang, A Filtered Bertini-type Theorem, J. reine angew. Math. 397 (1989), 214-219 Zbl0663.14008MR993224
  8. D. Eisenbud, Commutative algebra. With a view toward algebraic geometry, 150 (1995), Springer-Verlag, New York Zbl0819.13001MR1322960
  9. Ph. Ellia, M. Fiorentini, Défaut de postulation et singularités du Schéma de Hilbert, Annali Univ. di Ferrara 30 (1984), 185-198 Zbl0577.14019MR796926
  10. Ph. Ellia, A. Hirschowitz, E. Mezzetti, On the number of irreducible components of the Hilbert scheme of smooth space curves, International J. of Math. 3 (1992), 799-807 Zbl0824.14024MR1194072
  11. G. Ellingsrud, Sur le schéma de Hilbert des variétés de codimension 2 dans à cône de Cohen-Macaulay, Ann. Scient. Éc. Norm. Sup. 8 (1975), 423-432 Zbl0325.14002MR393020
  12. S. Ginouillac, Sur le nombre de composant du schéma de Hilbert des courbes ACM de , C. R. Acad. Sci. Paris Sér. I, Math. 329 (1999), 857-862 Zbl0961.14002MR1727997
  13. A. Grothendieck, Les schémas de Hilbert, (1960) 
  14. L. Gruson, Chr. Peskine, Genre des courbes de l’espace projectif, Proc. Tromsø 1977 687 (1978), Springer-Verlag, New York Zbl0412.14011MR527229
  15. S. Guffroy, Sur l’incomplétude de la série linéaire caractéristique d’une famille de courbes planes à nœuds et à cusps, Nagoya Math. J. 171 (2003), 51-83 Zbl1085.14501MR2002013
  16. R. Hartshorne, Algebraic Geometry, 52 (1983), Springer-Verlag, New York Zbl0531.14001MR360574
  17. A. Iarrobino, Betti strata of height two ideals, J. Algebra 285 (2005), 835-855 Zbl1095.13014MR2125467
  18. J. O. Kleppe, Deformations of graded algebras, Math. Scand. 45 (1979), 205-231 Zbl0436.14004MR580600
  19. J. O. Kleppe, Liaison of families of subschemes in , Algebraic Curves and Projective Geometry, Proceedings (Trento, 1988) 1389 (1989), Springer-Verlag, New York Zbl0697.14003MR1023396
  20. J. O. Kleppe, The Hilbert Scheme of Space Curves of small diameter, Annales de l’institut Fourier 56 (2006), 1297-1335 Zbl1117.14006MR2273858
  21. J. O. Kleppe, Families of Artinian and one-dimensional algebras, J. Algebra 311 (2007), 665-701 Zbl1129.14009MR2314729
  22. A. Laudal, Formal Moduli of Algebraic Structures, 754 (1979), Springer-Verlag, New York Zbl0438.14007MR551624
  23. M. Martin-Deschamps, D. Perrin, Sur la classification des courbes gauches, 184-185 (1990) Zbl0717.14017MR1073438
  24. M. Martin-Deschamps, D. Perrin, Courbes Gauches et Modules de Rao, J. reine angew. Math. 439 (1993), 103-145 Zbl0765.14017MR1219697
  25. J. Migliore, Introduction to liaison theory and deficiency modules, 165 (1998), Birkhäuser Boston, Inc., Boston, MA Zbl0921.14033MR1712469
  26. I. Peeva, Consecutive cancellations in Betti Numbers, Proc. Amer. Math. Soc. 132 (2004), 3503-3507 Zbl1099.13505MR2084070
  27. Ch. Peskine, L. Szpiro, Liaison des variétés algébrique, Invent. Math. 26 (1974), 271-302 Zbl0298.14022MR364271
  28. R. Piene, M. Schlessinger, On the Hilbert scheme compactification of the space of twisted cubics, Amer. J. Math. 107 (1985), 761-774 Zbl0589.14009MR796901
  29. A. Ragusa, G. Zappalá, On the reducibility of the postulation Hilbert scheme, Rend. Circ. Mat. Palermo, Serie II (2004), 401-406 Zbl1099.13029MR2165191
  30. A. P. Rao, Liaison Among Curves, Invent. Math. 50 (1979), 205-217 Zbl0406.14033MR520926
  31. E. Sernesi, Un esempio di curva ostruita in , (1981), 223-231 
  32. C. Walter, Some examples of obstructed curves in , Complex Projective Geometry 179 (1992) Zbl0787.14021MR1201393
  33. C. Walter, Transversality Theorems in General Characteristic with Applications to Arithmetically Buchsbaum Schemes, Internat. J. Math. 5 (1994), 609-617 Zbl0828.14031MR1284572
  34. C. A. Weibel, An introduction to homological algebra, 38 (1994), Cambridge University Press Zbl0797.18001MR1269324

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