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

Simis, Aron

doi:10.1090/S0002-9939-05-08169-4

The depth of the Jacobian ring of a homogeneous polynomial in three variables
HTML articles powered by AMS MathViewer

by Aron Simis

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

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

Published electronically: December 2, 2005

PDF | Request permission

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.

References

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).
David Eisenbud, Commutative algebra, Graduate Texts in Mathematics, vol. 150, Springer-Verlag, New York, 1995. With a view toward algebraic geometry. MR 1322960, DOI 10.1007/978-1-4612-5350-1
Francisco Calderón-Moreno and Luis Narváez-Macarro, The module $\scr Df^s$ for locally quasi-homogeneous free divisors, Compositio Math. 134 (2002), no. 1, 59–74. MR 1931962, DOI 10.1023/A:1020228824102
C. G. Gibson, Elementary geometry of algebraic curves: an undergraduate introduction, Cambridge University Press, Cambridge, 1998. MR 1663524, DOI 10.1017/CBO9781139173285
J. Herzog, A. Simis, and W. V. Vasconcelos, Koszul homology and blowing-up rings, Commutative algebra (Trento, 1981) Lecture Notes in Pure and Appl. Math., vol. 84, Dekker, New York, 1983, pp. 79–169. MR 686942
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
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.