The depth of the Jacobian ring of a homogeneous polynomial in three variables

Author:
Aron Simis

Journal:
Proc. Amer. Math. Soc. **134** (2006), 1591-1598

MSC (2000):
Primary 13C14, 13C15, 13H10, 13D02, 13D40, 13H15; Secondary 12E05, 14B05, 14H50

DOI:
https://doi.org/10.1090/S0002-9939-05-08169-4

Published electronically:
December 2, 2005

MathSciNet review:
2204268

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Abstract | References | Similar Articles | Additional Information

Abstract: The question as to whether the Jacobian ideal of an irreducible projective plane curve always admits an irrelevant component has been going around for some years. One shows that a curve will satisfy this if it has only ordinary nodes or cusps, while an example is given of a family of sextic curves whose respective Jacobian ideals are saturated. The connection between this problem and the theory of homogeneous free divisors in three variables is also pointed out, so the example gives a family of Koszul-free divisors.

**1.**D. Bayer and M. Stillman,*Macaulay*: a computer algebra system for algebraic geometry, Macaulay version 3.0 1994 (Macaulay for Windows by Bernd Johannes Wuebben, 1996).**2.**D. Eisenbud,*Commutative Algebra with a view toward Algebraic Geometry*, Springer-Verlag, Berlin, Heidelberg, New York, 1995. MR**1322960 (97a:13001)****3.**F. J. Calderón-Moreno and L. Narváez-Macarro, The module for locally quasi-homogeneous free divisors, Compos. Math.**134**, No.1, (2002), 59-74. MR**1931962 (2003i:14016)****4.**G. G. Gibson,*Elementary Geometry of Algebraic Curves: an Undergraduate Introduction*, Cambridge University Press, Cambridge, 1998. MR**1663524 (2000a:14002)****5.**J. Herzog, A. Simis, and W. Vasconcelos, Koszul homology and blowing-up rings, in COMMUTATIVE ALGEBRA, Proceedings: Trento 1981 (S. Greco and G. valla, Eds.), Lecture Notes in Pure and Applied Mathematics**84**, Marcel-Dekker, New York, 1983, pp. 79-169.MR**0686942 (84k:13015)****6.**K. Saito, Theory of logarithmic differential forms and logarithmic vector fields, J. Fac. Sci. Univ. Tokyo Sect. 1A Math.**27**(1980), 265-291. MR**0586450 (83h:32023)****7.**A. Simis, Differential idealizers and algebraic free divisors,*in*COMMUTATIVE ALGEBRA WITH A FOCUS ON GEOMETRIC AND HOMOLOGICAL ASPECTS, Proceedings of Sevilla, June 18-21, 2003 and Lisbon, June 23-27, 2003, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, vol. 244, 2005, pp. 211-226.

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Additional Information

**Aron Simis**

Affiliation:
Departamento de Matemática, CCEN, Universidade Federal de Pernambuco, Cidade Universitária, 50740-540 Recife, PE, Brazil

Email:
aron@dmat.ufpe.br

DOI:
https://doi.org/10.1090/S0002-9939-05-08169-4

Received by editor(s):
October 18, 2004

Received by editor(s) in revised form:
January 6, 2005

Published electronically:
December 2, 2005

Additional Notes:
The author was partially supported by a CNPq grant and the Brazil–France Cooperation in Mathematics.

Communicated by:
Bernd Ulrich

Article copyright:
© Copyright 2005
American Mathematical Society

The copyright for this article reverts to public domain 28 years after publication.