Abstract
Following the path trodden by several authors along the border between Algebraic Geometry and Algebraic Combinatorics, we present some new results on the combinatorial structure of Borel ideals. This enables us to prove theorems on the shape of thesectional matrix of a homogeneous ideal, which is a new invariant stronger than the Hilbert function.
Similar content being viewed by others
References
[AHH97] A. Aramova, J. Herzog, and T. Hibi, Gotzmann theorems for exterior algebras and combinatorics, J. Algebra191 (1997) 174–211.
[AM69] M.F. Atiyah and J.G. MacDonald, Introduction to Commutative Algebra, Addison-Wesley, Reading, Massachusetts, 1969.
[Bay82] D. Bayer, The Division Algorithm and the Hiibert Scheme, PhD thesis, Harvard University, Microfilms International, 300 N, Zeeb Rd., Ann Arbor, MI 48106, 1982, Order-Nr, 82-22588.
[Big93] A.M. Bigatti, Upper bounds for the Betti numbers of a given Hilbert function, Comm. Algebra21 (7) (1993) 2317–2334.
[Big95] A.M. Bigatti, Aspetti Combinatorici e Computazionali dell'Algebra Commutativa, PhD thesis, Università di Torino, 1995.
[BS87a] D. Bayer and M. Stillman, A criterion for detectingm-regularity, Invent. Math.87 (1987) 1–11.
[BS87b] D. Bayer and M. Stillman, A theorem on refining division orders by the reverse lexicographic order, Duke Math. J.55 (1987) 321–328.
[EK90] S. Eliahou and M. Kervaire, Minimal resolution of some monomial ideals, J. Algebra129 (1990) 1–25.
[Eli97] J. Elias, Sharp upper bound for the Betti numbers of Cohen-Macaulay modules, Illinois J. Math.41 (3) (1997) 1–27.
[Gal74] A. Galligo, A propos du théorème de préparation de Weierstrass, functions de plusieurs variables complexes, Lecture Notes in Mathematics, Vol. 409, 1974, pp. 543–579.
[Gas97] V. Gasharov, Extremal properties of Hilbert functions, Illinois J. Math. (1997), to appear.
[Got78] G. Gotzmann, Eine Bedingung für die Flachheit und das Hilbertpolynom eines graduierten Ringes, Math. Z.158 (1978) 61–70.
[Gre88] M. Green, Restriction of linear series to hyperplanes, and some results of Macaulay and Gotzmann, algebraic curves and projective geometry, In: Proceedings, Trento, Lecture Notes in Mathematics, Vol. 1389, Springer, 1988.
[Gre96] M. Green, Summer School on Commutative Algebra, Vol. 2 of Quaderns, Chapter Generic initial ideals, pp. 11–85, Centre de Recerca Matematica, July 1996.
[HP97] J. Herzog and D. Popescu, Hilbert functions and generic forms, Compositio Math. (1997), to appear.
[Hul93] H. Hulett, Maximum Betti numbers of homogeneous ideals with a given Hilbert function, Comm. Algebra21 (7) (1993) 2335–2350.
[Mac27] F.S. Macaulay, Some properties of enumeration in the theory of modular systems, Proc. London Math. Soc.26 (1927) 531–555.
[Mal96] D. Mall, On the Castelnuovo-Mumford regularity, 1996.
[Mal97] D. Mall, Characterizations of lexicographic sets and simply-connected Hilbert schemes, In: Proceedings of AAECC-12, Lecture Notes in Computer Science, Vol. 1255, Springer Verlag, 1997, pp. 221–136.
[MV96] M. Miller and R.H. Villarreal, A note on generators of least degree in Gorenstein ideals, Proc. Amer. Math. Soc.124 (2) (1996).
[Rob90] L. Robbiano, Introduction to the theory of Hilbert function, Queen's Papers in Pure and Applied Mathematics85 (1990) B1-B26.
Author information
Authors and Affiliations
Additional information
The authors were partially supported by the Consiglio Nazionale delle Ricerche (CNR).
Rights and permissions
About this article
Cite this article
Bigatti, A.M., Robbiano, L. Borel sets and sectional matrices. Annals of Combinatorics 1, 197–213 (1997). https://doi.org/10.1007/BF02558475
Received:
Issue Date:
DOI: https://doi.org/10.1007/BF02558475