Regularity bounds by minimal generators and Hilbert function

• Autores: F. Cioffi, M. G. Marinari, Luciana Ramella
• Localización: Collectanea mathematica, ISSN 0010-0757, Vol. 60, Fasc. 1, 2009, págs. 89-100
• Idioma: español
• DOI: 10.1007/bf03191218
• Enlaces
  • Texto completo
• Resumen

  • Let $\rho_C$ be the regularity of the Hilbert function of a projective curve $C$ in ${\bf P}^n_K$ over an algebraically closed field $K$ and $\beta_1,\ldots,\beta_{n-1}$ be degrees for which there exists a complete intersection of type $(\beta_1,\ldots,\beta_{n-1})$ containing properly $C$. Then the Castelnuovo-Mumford regularity of $C$ is upper bounded by $\max\{\rho_C+1 \beta_1+ \ldots+ \beta_{n-1}-(n-1)\}$. We investigate the sharpness of the above bound, which is achieved by curves algebraically linked to ones having degenerate general hyperplane section.

• Referencias bibliográficas
  • E. Ballico, On singular curves in the case of positive characteristic, Math. Nachr. 141 (1989), 267–273.
  • D. Bayer and M. Stillman, A criterion for detecting m-regularity, Invent. Math. 87 (1987), 1–11.
  • A. Bertram, L. Ein, and R. Lazarsfeld, Vanishing theorems, a theorem of Severi, and the equations defining projective varieties, J. Amer....
  • M. Brodmann and U. Nagel, Bounding cohomological Hilbert functions by hyperplane sections, J. Algebra 174 (1995), 323–348.
  • M. Chardin, A. Fall, and U. Nagel, Bounds for the Castelnuovo-Mumford regularity of modules, Math. Z. 258 (2008), 69–80.
  • M. Chardin and B. Ulrich, Liaison and Castelnuovo-Mumford regularity, Amer. J. Math. 124 (2002), 1103–1124.
  • CoCoA Team, CoCoA: a system for doing Computations in Commutative Algebra, Available at http://cocoa.dima.unige.it.
  • E.D. Davis, A.V. Geramita, and F. Orecchia, Gorenstein algebras and the Cayley-Bacharach theorem, Proc. Amer. Math. Soc. 93 (1985), 593–597.
  • M.L. Green, Generic initial ideals, Six Lectures on Commutative Algebra (Bellaterra, 1996), 119–186, Progr. Math. 166, Birkhäuser, Basel,...
  • J. Harris, The genus of space curves, Math. Ann. 249 (1980), 191–204.
  • R. Hartshorne, Algebraic Geometry, Graduate Texts in Mathematics 52, Springer-Verlag, New York-Heidelberg, 1977.
  • R. Hartshorne, The genus of space curves, Ann. Univ. Ferrara Sez. VII (N.S.) 40 (1994–1996), 207–223.
  • E. Kunz, Introduction to Commutative Algebra and Algebraic Geometry, Birkhäuser Boston Inc., Boston, MA, 1985.
  • R. Maggioni and A. Ragusa, The Hilbert function of generic plane sections of curves of ℙ³, Invent. Math. 91 (1988), 253–258.
  • H.M. Martin, J.C. Migliore, and S. Nollet, Degrees of generators of ideals defining curves in projective space, Comm. Algebra 26 (1998), 1209–1231.
  • J.C. Migliore, Introduction to Liaison Theory and Deficiency Modules, Progress in Mathematics 165, Birkhäuser Boston Inc., Boston, MA, 1998.
  • J.C. Migliore and Rosa M. Miró-Roig, On k-Buchsbaum curves in ℙ³, Comm. Algebra 18 (1990), 2403–2422.
  • Ch. Miyazaki, Sharp bounds on Castelnuovo-Mumford regularity, Trans. Amer. Math. Soc. 352 (2000), 1675–1686.
  • U. Nagel and P. Schenzel, Degree bounds for generators of cohomology modules and Castelnuovo-Mumford regularity, Nagoya Math. J. 152 (1998),...
  • S. Nollet, Integral subschemes of codimension two, J. Pure Appl. Algebra 141 (1999), 269–288.
  • F. Orecchia, F. Cioffi, and I. Ramella, Points (software for computations on points), Available for Linux platform at http://wpage.unina.it/cioffifr/EPoints.html,...
  • R. Strano, On generalized Laudal's lemma, Complex Projective Geometry (Trieste, 1989/Bergen, 1989), 284–293, London Math. Soc. Lecture...