Cohen-Macaulay Criteria for Projective Monomial Curves via Gröbner Bases

Abstract

We prove new characterizations based on Gröbner bases for the Cohen-Macaulay property of a projective monomial curve.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Arslan, F.: Cohen-Macaulayness of tangent cones. Proc. Am. Math. Soc. 128(8), 2243–2251 (2000)

    MathSciNet  Article  MATH  Google Scholar 

  2. 2.

    Arslan, F., Mete, P., Şahin, M.: Gluing and Hilbert functions of monomial curves. Proc. Am. Math. Soc. 137(7), 2225–2232 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  3. 3.

    Bresinsky, H.: On prime ideals with generic zero \(x_{i}=t^{n_{i}}\). Proc. Am. Math. Soc. 47, 329–332 (1975)

    MATH  Google Scholar 

  4. 4.

    Bresinsky, H., Schenzel, P., Vogel, W.: On liaison, arithmetical Buchsbaum curves and monomial curves in \(\mathbb {P}^{3}\). J. Algebra 86(2), 283–301 (1984)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Cavaliere, M. P., Niesi, G.: On monomial curves and Cohen-Macaulay type. Manuscripta Math. 42(2–3), 147–159 (1983)

    MathSciNet  Article  MATH  Google Scholar 

  6. 6.

    Cimpoeaş, M., Stamate, D.I.: Gröbner–nice pairs of ideals. Preprint (2018)

  7. 7.

    Cox, D., Little, J., O’Shea, D.: Ideals, Varieties, and Algorithms, 3rd edn. Springer, New York (2007)

  8. 8.

    Decker, W., Greuel, G.-M., Pfister, G., Schönemann, H.: Singular 3-1-6 — a computer algebra system for polynomial computations. http://www.singular.uni-kl.de (2012)

  9. 9.

    Eisenbud, D.: Commutative Algebra. With a View Toward Algebraic Geometry Graduate Texts in Mathematics, vol. 150. Springer, New York (1995)

    Google Scholar 

  10. 10.

    Ene, V., Herzog, J.: Gröbner Bases in Commutative Algebra Graduate Studies in Mathematics, vol. 130. American Medical Association, Providence (2012)

    Google Scholar 

  11. 11.

    Goto, S., Suzuki, N., Watanabe, K.-I.: On affine semigroup rings. Japan J. Math. (N.S.) 2(1), 1–12 (1976)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Herzog, J.: Generators and relations of Abelian semigroups and semigroup rings. Manuscripta Math. 3, 175–193 (1970)

    MathSciNet  Article  MATH  Google Scholar 

  13. 13.

    Herzog, J., Hibi, T.: Monomial Ideals Graduate Texts in Mathematics, vol. 260. Springer, London (2011)

    Google Scholar 

  14. 14.

    Herzog, J., Stamate, D.I.: On the defining equations of the tangent cone of a numerical semigroup ring. J. Algebra 418, 8–28 (2014)

    MathSciNet  Article  MATH  Google Scholar 

  15. 15.

    Kamoi, Y.: Defining ideals of Cohen-Macaulay semigroup rings. Commun. Algo. 20(11), 3163–3189 (1992)

    MathSciNet  Article  MATH  Google Scholar 

  16. 16.

    Kamoi, Y.: Defining ideals of Buchsbaum semigroup rings. Nagoya Math. J. 136, 115–131 (1994)

    MathSciNet  Article  MATH  Google Scholar 

  17. 17.

    Morales, M., Dung, N.T.: Gröbner basis, a “pseudo-polynomial” algorithm for computing the Frobenius number. Preprint. arXiv:1510.01973 [math.AC] (2015)

  18. 18.

    Roune, B. H.: Solving thousand-digit Frobenius problems using Gröbner bases. J. Symb. Comput. 43(1), 1–7 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  19. 19.

    Rosales, J. C., García-Sánchez, P. A., Urbano-Blanco, J. M.: On Cohen-Macaulay subsemigroups of \(\mathbb {N}^{2}\). Comm. Algebra 26(8), 2543–2558 (1998)

    MathSciNet  Article  MATH  Google Scholar 

  20. 20.

    Vu, T.: Periodicity of Betti numbers of monomial curves. J. Algebra 418, 66–90 (2014)

    MathSciNet  Article  MATH  Google Scholar 

Download references

Acknowledgements

We gratefully acknowledge the use of the Singular [8] software for our computations.

Funding

Dumitru Stamate was supported by the University of Bucharest, Faculty of Mathematics and Computer Science, through the 2017 Mobility Fund.

Author information

Affiliations

Authors

Corresponding author

Correspondence to Jürgen Herzog.

Additional information

Publisher’s Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Herzog, J., Stamate, D.I. Cohen-Macaulay Criteria for Projective Monomial Curves via Gröbner Bases. Acta Math Vietnam 44, 51–64 (2019). https://doi.org/10.1007/s40306-018-00302-5

Download citation

Keywords

  • Arithmetically Cohen-Macaulay
  • Projective monomial curve
  • Revlex
  • Gröbner basis
  • Numerical semigroup
  • Apéry set

Mathematics Subject Classification (2010)

  • Primary 13H10
  • 13P10
  • 16S36
  • Secondary 13F20
  • 14M25