Disproving Hibi’s Conjecture with CoCoA or Projective Curves with bad Hilbert Functions

  • Gianfranco Niesi
  • Lorenzo Robbiano
Part of the Progress in Mathematics book series (PM, volume 109)


In this paper we show how to combine different techniques from Commutative Algebra and a systematic use of a Computer Algebra System (in our case mainly CoCoA (see [G-N] and [A-G-N])) in order to explicitly construct Cohen-Macaulay domains, which are standard k-algebras and whose Hilbert function is “bad”. In particular we disprove a well-known conjecture by Hibi.


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. [A-G-N]
    Armando, E., Giovini, A., Niesi, G., CoCoA User’s Manual v. 1.5, (1991), Dipartimento di Matematica, Università di Genova.Google Scholar
  2. [A-M]
    Atiyah, M. F., Macdonald, I. G., Introduction to Commutative Algebra, Addison-Wesley (1969).Google Scholar
  3. [B-C-R]
    Bigatti, A., Caboara, M., Robbiano, L., On the computation of Hilbert-Poincaré series, Applicable Algebra in Engineering, Communications and Computing (1990), To appear.Google Scholar
  4. [G]
    Galligo, A., Exemples d’ensembles de Points en Position Uniforme, Proceedings of MEGA-90 (1991), Birkhauser.Google Scholar
  5. [G-K-R]
    Geramita, A., Kreuzer, M., Robbiano, L., Cayley-Bacharach schemes and their canonical modules, Trans. Amer. Math. J. (1991), To appear.Google Scholar
  6. [G-M-R]
    Geramita, A.V., Maroscia, P., Roberts, L.G., The Hilbert function of a reduced k-algebra, J. London Math. Soc. 28 (1983), 443–452.CrossRefzbMATHMathSciNetGoogle Scholar
  7. [G-M]
    Geramita, A.V., Migliore, J.C., Hyperplane sections of a smooth curve in3, Comm. Algebra 17 (1989), 3129–3164.CrossRefzbMATHMathSciNetGoogle Scholar
  8. [G-N]
    Giovini, A., Niesi, G., CoCoA: a user-friendly system for commutative algebra, Proceedings of DISCO-90, Lecture Notes in Computer Sciences 429 (1990), Springer-Verlag.Google Scholar
  9. [Ha]
    Harris, J., The genus of space curves, Math. Ann. 249 (1980), 191–204.CrossRefzbMATHMathSciNetGoogle Scholar
  10. [H-E]
    Harris, J. (with the collaboration of D. Eisenbud), Curves in projective space in Sém. de Mathématiques Supérieures, Université de Montreal (1982),.Google Scholar
  11. [Hart]
    Hartshorne, R., Algebraic Geometry, Springer (1977).Google Scholar
  12. [Hi]
    Hibi, T., Flawless O-sequences and Hilbert functions of Cohen-Macaulay integral domains, J. Pure Appl. Algebra 60 (1989), 245–251.CrossRefzbMATHMathSciNetGoogle Scholar
  13. [Ra]
    Rathmann, J., The uniform position principle for curves in characteristic p, Math. Ann. 276 (1987), 565–579.CrossRefzbMATHMathSciNetGoogle Scholar
  14. [Ri]
    Robbiano, L., Introduction to the theory of Gröbner bases in The Curves Seminar at Queen’s, vol. V, Queen’s Papers in Pure and Appl. Math. 80 (1988), Queen’s University, Kingston B1–B29.Google Scholar
  15. [R2]_Robbiano, L., On the theory of Hilbert functions, Queen’s Papers in Pure and Appl. Math., Kingston Canada 85 (1990), Vol VII.Google Scholar
  16. [S-S]
    Scheja, G., Storch, U., Lehrbuch der Algebra, B. G. Teubner Stuttgart (1988).Google Scholar
  17. [S1]_Stanley, R., Hilbert functions of graded algebras, Adv. in Math. 28 (1978), 57–83.CrossRefzbMATHGoogle Scholar
  18. [S2]_Stanley, R., On the Hilbert function of a graded Cohen-Macaulay domain, (1990), preprint, Massachusetts Inst. of Technology, Cambridge.Google Scholar
  19. [W]
    van der Waerden, B.L., Algebra, Vol I, Ungar (1970).Google Scholar

Copyright information

© Birkhäuser Boston 1993

Authors and Affiliations

  • Gianfranco Niesi
    • 1
  • Lorenzo Robbiano
    • 1
  1. 1.Dipartimento di Matematica dell’Universitá di GenovaGenovaItaly

Personalised recommendations