CohenMacaulayness of tangent cones
Author:
Feza Arslan
Journal:
Proc. Amer. Math. Soc. 128 (2000), 22432251
MSC (1991):
Primary 14H20; Secondary 13H10, 13P10
Published electronically:
November 29, 1999
MathSciNet review:
1653409
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Abstract: We give a criterion for checking the CohenMacaulayness 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 CohenMacaulay tangent cone of a monomial curve in an affine space can be arbitrarily large for contrary to the 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.
 1.
Dave
Bayer and Mike
Stillman, Computation of Hilbert functions, J. Symbolic
Comput. 14 (1992), no. 1, 31–50. MR 1177988
(94f:13018), http://dx.doi.org/10.1016/07477171(92)90024X
 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.
H.
Bresinsky, On prime ideals with generic zero
𝑥ᵢ=𝑡^{𝑛ᵢ}, Proc. Amer. Math. Soc. 47 (1975), 329–332. MR 0389912
(52 #10741), http://dx.doi.org/10.1090/S00029939197503899120
 4.
Maria
Pia Cavaliere and Gianfranco
Niesi, On form ring of a onedimensional semigroup ring,
Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math.,
vol. 84, Dekker, New York, 1983, pp. 39–48. MR 686939
(84i:13018)
 5.
David
Cox, John
Little, and Donal
O’Shea, Ideals, varieties, and algorithms, Undergraduate
Texts in Mathematics, SpringerVerlag, New York, 1992. An introduction to
computational algebraic geometry and commutative algebra. MR 1189133
(93j:13031)
 6.
Arnaldo
García, CohenMacaulayness of the associated graded of a
semigroup ring, Comm. Algebra 10 (1982), no. 4,
393–415. MR
649344 (83k:13013), http://dx.doi.org/10.1080/00927878208822724
 7.
Jürgen
Herzog, Generators and relations of abelian semigroups and
semigroup rings., Manuscripta Math. 3 (1970),
175–193. MR 0269762
(42 #4657)
 8.
J.
Herzog, When is a regular sequence super regular?, Nagoya
Math. J. 83 (1981), 183–195. MR 632652
(83c:13009)
 9.
Marcel
Morales, Noetherian symbolic blowups, J. Algebra
140 (1991), no. 1, 12–25. MR 1114901
(92c:13020), http://dx.doi.org/10.1016/00218693(91)90141T
 10.
Lorenzo
Robbiano and Giuseppe
Valla, On the equations defining tangent cones, Math. Proc.
Cambridge Philos. Soc. 88 (1980), no. 2,
281–297. MR
578272 (81i:14004), http://dx.doi.org/10.1017/S0305004100057583
 11.
Judith
D. Sally, Good embedding dimensions for Gorenstein
singularities, Math. Ann. 249 (1980), no. 2,
95–106. MR
578716 (82c:13031), http://dx.doi.org/10.1007/BF01351406
 12.
Wolmer
V. Vasconcelos, Computational methods in commutative algebra and
algebraic geometry, Algorithms and Computation in Mathematics,
vol. 2, SpringerVerlag, Berlin, 1998. With chapters by David
Eisenbud, Daniel R. Grayson, Jürgen Herzog and Michael Stillman. MR 1484973
(99c:13048)
 1.
 Bayer, D., Stillman, M., Computation of Hilbert functions, J. Symbolic Computation 14 (1992), 3150. 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 , Proc. Amer. Math. Soc. 47, No.2 (1975), 329332. MR 52:10741
 4.
 Cavaliere, M. P., Niesi, G., On form ring of a onedimensional semigroup ring, Lecture Notes in Pure and Appl. Math. 84 (1983), 3948. MR 84i:13018
 5.
 Cox, D., Little, J., O'Shea, D., Ideals, varieties and algorithms, SpringerVerlag, 1992. MR 93j:13031
 6.
 Garcia, A., CohenMacaulayness of the associated graded of a semigroup ring, Comm. in Algebra 10, No.4 (1982), 393415. MR 83k:13013
 7.
 Herzog, J., Generators and relations of abelian semigroups and semigroup rings, Manuscripta Math. 3 (1970), 175193. MR 42:4657
 8.
 Herzog, J., When is a regular sequence super regular?, Nagoya Math. J., 83 (1981), 183195. MR 83c:13009
 9.
 Morales, M., Noetherian symbolic blowups, Journal of Algebra 140 (1991), 1225. MR 92c:13020
 10.
 Robbiano, L., Valla, G., On the equations defining tangent cones, Math. Proc. Camb. Phil. Soc. 88 (1980), 281297. MR 81i:14004
 11.
 Sally, J., Good embedding dimensions for Gorenstein singularities, Math. Ann. 249 (1980), 95106. MR 82c:13031
 12.
 Vasconcelos, W. V., Computational methods in commutative algebra and algebraic geometry, SpringerVerlag, 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:
http://dx.doi.org/10.1090/S0002993999052296
PII:
S 00029939(99)052296
Keywords:
CohenMacaulay 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
