The depth of the Jacobian ring of a homogeneous polynomial in three variables
Author:
Aron Simis
Journal:
Proc. Amer. Math. Soc. 134 (2006), 15911598
MSC (2000):
Primary 13C14, 13C15, 13H10, 13D02, 13D40, 13H15; Secondary 12E05, 14B05, 14H50
Published electronically:
December 2, 2005
MathSciNet review:
2204268
Fulltext PDF Free Access
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 Koszulfree 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.
David
Eisenbud, Commutative algebra, Graduate Texts in Mathematics,
vol. 150, SpringerVerlag, New York, 1995. With a view toward
algebraic geometry. MR 1322960
(97a:13001)
 3.
Francisco
CalderónMoreno and Luis
NarváezMacarro, The module 𝒟𝒻^{𝓈}
for locally quasihomogeneous free divisors, Compositio Math.
134 (2002), no. 1, 59–74. MR 1931962
(2003i:14016), http://dx.doi.org/10.1023/A:1020228824102
 4.
C.
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.
V. Vasconcelos, Koszul homology and blowingup rings,
Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math.,
vol. 84, Dekker, New York, 1983, pp. 79–169. MR 686942
(84k:13015)
 6.
Kyoji
Saito, Theory of logarithmic differential forms and logarithmic
vector fields, J. Fac. Sci. Univ. Tokyo Sect. IA Math.
27 (1980), no. 2, 265–291. MR 586450
(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 1821, 2003 and Lisbon, June 2327, 2003, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, vol. 244, 2005, pp. 211226.
 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, SpringerVerlag, Berlin, Heidelberg, New York, 1995. MR 1322960 (97a:13001)
 3.
 F. J. CalderónMoreno and L. NarváezMacarro, The module for locally quasihomogeneous free divisors, Compos. Math. 134, No.1, (2002), 5974. 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 blowingup rings, in COMMUTATIVE ALGEBRA, Proceedings: Trento 1981 (S. Greco and G. valla, Eds.), Lecture Notes in Pure and Applied Mathematics 84, MarcelDekker, New York, 1983, pp. 79169.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), 265291. 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 1821, 2003 and Lisbon, June 2327, 2003, Lecture Notes in Pure and Applied Mathematics, Marcel Dekker, vol. 244, 2005, pp. 211226.
Similar Articles
Retrieve articles in Proceedings of the American Mathematical Society
with MSC (2000):
13C14,
13C15,
13H10,
13D02,
13D40,
13H15,
12E05,
14B05,
14H50
Retrieve articles in all journals
with MSC (2000):
13C14,
13C15,
13H10,
13D02,
13D40,
13H15,
12E05,
14B05,
14H50
Additional Information
Aron Simis
Affiliation:
Departamento de Matemática, CCEN, Universidade Federal de Pernambuco, Cidade Universitária, 50740540 Recife, PE, Brazil
Email:
aron@dmat.ufpe.br
DOI:
http://dx.doi.org/10.1090/S0002993905081694
PII:
S 00029939(05)081694
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.
