The Hilbert schemes of locally Cohen–Macaulay curves in ℙ3 may after all be connected

• Autores: Paolo Lella, Enrico Schlesinger
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 64, Fasc. 3, 2013, págs. 363-372
• Idioma: inglés
• DOI: 10.1007/s13348-012-0062-3
• Texto completo no disponible (Saber más ...)
• Resumen

  • Progress on the problem whether the Hilbert schemes of locally Cohen–Macaulay curves in ℙ3 are connected has been hampered by the lack of an answer to a question raised by Robin Hartshorne in (Commun. Algebra 28:6059–6077, 2000) and more recently in (American Institute of Mathematics, Workshop components of Hilbert schemes, problem list, 2010. http://aimpl.org/hilbertschemes): does there exist a flat irreducible family of curves whose general member is a union of d ≥ 4 disjoint lines on a smooth quadric surface and whose special member is a locally Cohen–Macaulay curve in a double plane? In this paper we give a positive answer to this question: for every d we construct a family with the required properties, whose special fiber is an extremal curve in the sense by Martin-Deschamps and Perrin (Ann. Sci. E.N.S. 4 e Série 29:757–785, 1996). From this we conclude that every effective divisor in a smooth quadric surface is in the connected component of its Hilbert scheme that contains extremal curves.

• Referencias bibliográficas
  • Aït Amrane S.: Sur le schéma de Hilbert des courbes de degré d et genre (d−3)(d−4)/2 de P 3 k . C R Acad. Sci. Paris Sér. I Math. 326(7),...
  • American Institute of Mathematics problem list. Workhop components of Hilbert schemes (2010). http://aimpl.org/hilbertschemes
  • Ballico E., Bolondi G., Migliore J.: The Lazarsfeld–Rao problem for liaison classes of two-codimensional subschemes of {\mathbb{P}^n} . Am....
  • Bayer, D., Mumford, D.: What can be computed in algebraic geometry? In: Computational algebraic geometry and commutative algebra (Cortona,...
  • Eisenbud, D.: Commutative algebra. GTM150, Springer, New York, p. xvi+785 (1995)
  • Grayson, D., Stillman, M.: Macaulay2, a software system for research in algebraic geometry. http://www.math.uiuc.edu/Macaulay2/
  • Hartshorne R.: Connectedness of the Hilbert scheme. Inst. Hautes études Sci. Publ. Math. 29, 5–48 (1966)
  • Hartshorne R.: Algebraic Geometry. Springer, New York (1977)
  • Hartshorne R.: Generalized divisors on Gorenstein schemes. K-Theory 8(3), 287–339 (1994)
  • Hartshorne, R.: On the connectedness of the Hilbert scheme of curves in {\mathbb{P}^{3}} . Commun. Algebra 28(12), 6059–6077 (2000) (special...
  • Hartshorne R.: On Rao’s theorems and the Lazarsfeld–Rao property. Ann. Fac. Sci. Toulouse Math. 6(3), 375–393 (2003)
  • Hartshorne, R.: Questions of connectedness of the Hilbert scheme of curves in {\mathbb{P}^{3}}. In: Algebra, arithmetic and geometry with...
  • Hartshorne R., Martin-Deschamps M., Perrin D.: Triades et familles de courbes gauches. Math. Ann. 315(3), 397–468 (1999)
  • Hartshorne, R., Schlesinger, E.: Curves in the double plane. Commun. Algebra 28(12), 5655–5676 (2000) (special issue in honor of Robin Hartshorne)
  • Lazarsfeld, R., Rao, A.P.: Linkage of general curves of large degree. In: Algebraic geometry open problems (Ravello, 1982). Lecture Notes...
  • Martin-Deschamps, M., Perrin, D.: Sur la classification des courbes gauches. Astérisque, pp. 184–185. Société Mathématique de France (1990)
  • Martin-Deschamps M., Perrin D.: Sur les bornes du module de Rao. C.R. Acad. Sci. Paris Série I 137, 1159–1162 (1993)
  • Martin-Deschamps M., Perrin D.: Le schéma de Hilbert des Courbes Gauches localement Cohen–Macaulay nést (presque) jamais réduit. Ann. Sci....
  • Migliore, J.C.: Introduction to Liaison Theory and Deficiency Modules. Birkhäuser, Boston (1998)
  • Nollet S.: The Hilbert schemes of degree three curves. Ann. Sci. école Norm. Sup. 30(3), 367–384 (1997)
  • Nollet S.: Subextremal curves. Manuscr. Math. 94(3), 303–317 (1997)
  • Nollet S., Schlesinger E.: Hilbert schemes of degree four curves. Compos. Math. 139(2), 169–196 (2003)
  • Perrin D.: Un pas vers la connexité du schéma de Hilbert: les courbes de Koszul sont dans la composante des extrémales. Collect. Math. 52(3),...
  • Rao A.P.: Liaison among curves in {\mathbb{P}^{3}}. Invent. Math. 50(3), 205–217 (1979)
  • Sabadini, I.: On the Hilbert scheme of curves of degree d and genus {\binom{d-3}{2}-1}. Le Matematiche 45, 517–531 (2000) (volume dedicated...
  • Schlesinger, E.: Footnote to a paper by R. Hartshorne: On the connectedness of the Hilbert scheme of curves in {\mathbb{P}^{3}}. Commun. Algebra...
  • Strano R.: Biliaison classes of curves in {\mathbb{P}^{3}}. Proc. Am. Math. Soc. 132(3), 649–658 (2004)