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Cohen-Macaulayness of tangent cones


Author: Feza Arslan
Journal: Proc. Amer. Math. Soc. 128 (2000), 2243-2251
MSC (1991): Primary 14H20; Secondary 13H10, 13P10
DOI: https://doi.org/10.1090/S0002-9939-99-05229-6
Published electronically: November 29, 1999
MathSciNet review: 1653409
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Abstract: We give a criterion for checking the Cohen-Macaulayness of the tangent cone of a monomial curve by using the Gröbner basis. For a family of monomial curves, we give the full description of the defining ideal of the curve and its tangent cone at the origin. By using this family of curves and their extended versions to higher dimensions, we prove that the minimal number of generators of a Cohen-Macaulay tangent cone of a monomial curve in an affine $l$-space can be arbitrarily large for $l \geq 4$ contrary to the $l=3$ case shown by Robbiano and Valla. We also determine the Hilbert series of the associated graded ring of this family of curves and their extended versions.


References [Enhancements On Off] (What's this?)

  • 1. Bayer, D., Stillman, M., Computation of Hilbert functions, J. Symbolic Computation 14 (1992), 31-50. MR 94f:13018
  • 2. D. Bayer and M. Stillman, Macaulay, A system for computation in algebraic geometry and commutative algebra, 1992, available via anonymous ftp from math.harvard.edu.
  • 3. Bresinsky, H., On prime ideals with generic zero $x_{i}=t^{n_{i}}$, Proc. Amer. Math. Soc. 47, No.2 (1975), 329-332. MR 52:10741
  • 4. Cavaliere, M. P., Niesi, G., On form ring of a one-dimensional semigroup ring, Lecture Notes in Pure and Appl. Math. 84 (1983), 39-48. MR 84i:13018
  • 5. Cox, D., Little, J., O'Shea, D., Ideals, varieties and algorithms, Springer-Verlag, 1992. MR 93j:13031
  • 6. Garcia, A., Cohen-Macaulayness of the associated graded of a semigroup ring, Comm. in Algebra 10, No.4 (1982), 393-415. MR 83k:13013
  • 7. Herzog, J., Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 (1970), 175-193. MR 42:4657
  • 8. Herzog, J., When is a regular sequence super regular?, Nagoya Math. J., 83 (1981), 183-195. MR 83c:13009
  • 9. Morales, M., Noetherian symbolic blow-ups, Journal of Algebra 140 (1991), 12-25. MR 92c:13020
  • 10. Robbiano, L., Valla, G., On the equations defining tangent cones, Math. Proc. Camb. Phil. Soc. 88 (1980), 281-297. MR 81i:14004
  • 11. Sally, J., Good embedding dimensions for Gorenstein singularities, Math. Ann. 249 (1980), 95-106. MR 82c:13031
  • 12. Vasconcelos, W. V., Computational methods in commutative algebra and algebraic geometry, Springer-Verlag, 1998. MR 99c:13048

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Additional Information

Feza Arslan
Affiliation: Department of Mathematics, Bilkent University, Ankara, Turkey 06533
Address at time of publication: Department of Mathematics, METU, Ankara, Turkey 06531
Email: sarslan@fen.bilkent.edu.tr, feza@arf.math.metu.edu.tr

DOI: https://doi.org/10.1090/S0002-9939-99-05229-6
Keywords: Cohen-Macaulay ring, monomial curve, tangent cone
Received by editor(s): May 1, 1998
Received by editor(s) in revised form: September 18, 1998
Published electronically: November 29, 1999
Additional Notes: The author was supported by TÜBİTAK BDP Grant.
Communicated by: Wolmer V. Vasconcelos
Article copyright: © Copyright 2000 American Mathematical Society

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