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Self-linked curve singularities

Published online by Cambridge University Press:  22 January 2016

Jürgen Herzog  and
Bernd Ulrich
Jürgen Herzog
Affiliation:
Fachbereich Mathematik, Universität-Gesamthochschule-Essen, Universitätstrasse 3, 4300 Essen, Germany
Bernd Ulrich
Affiliation:
Department of Mathematics, Michigan State University, East Lansing, Michigan 48824, USA
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Let S be a three-dimensional regular local ring and let I be a prime ideal in S of height two. This paper is motivated by the question of when I is a set-theoretic complete intersection and when the symbolic Rees algebra S(I) = ⊕n≥0I(n)tn is Noetherian. The connection between the two problems is given by a result of Cowsik which says that the Noetherian property of S(I) implies that I is a set-theoretic complete intersection ([1]).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 120 , December 1990 , pp. 129 - 153
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1990

References

[1] Cowsik, R., Symbolic powers and number of defining equations, in “Algebra and its applications (New Delhi, 1981)”, Lecture Notes in Pure and Appl. Math., Vol. 91, Dekker, New York, 1984, 13–14.Google Scholar
[2] Cowsik, R. and Nori, , On the fibres of blowing-up, J. Indian Math. Soc., 40 (1976), 217–222.Google Scholar
[3] Eliahou, S., Symbolic powers of monomial curves, preprint.Google Scholar
[4] Hartshorne, R., Complete intersections and connectedness, Amer. J. Math., 84 (1962), 497–508.Google Scholar
[5] Herzog, J., Generators and relations of abelian semigroups and semigroups rings, Manuscripta Math., 3 (1970), 153–193.Google Scholar
[6] Herzog, J., Certain complexes associated to a sequence and a matrix, Manuscripta Math., 12 (1974), 217–247.Google Scholar
[7] Herzog, J., Ein Cohen-Macaulay-Kriterium mit Anwendungen auf den Konormalen-modul und den Differentialmodul, Math. Z., 163 (1978), 149–162.CrossRefGoogle Scholar
[8] Herzog, J., Simis, A., and Vasconcelos, W., Koszul homology and blowing-up rings, in “Commutative algebra, Proc. Trento Conf.”, eds.: Greco, S. and Valla, G., Lecture Notes in Pure and Appl. Math., Vol. 84, Dekker, New York, 1983, 79–169.Google Scholar
[9] Huneke, C., On the finite generation of symbolic blow-ups, Math. Z., 179 (1982), 465–472.CrossRefGoogle Scholar
[10] Huneke, C., The primary components of and integral closures of ideals in 3-dimensional regular local rings, Math. Ann., 275 (1986), 617–635.Google Scholar
[11] Huneke, C., Hilbert functions and symbolic powers, Michigan Math. J., 34 (1987), 293–318.Google Scholar
[12] Huneke, C. and Ulrich, B., The structure of linkage, Annals of Math., 126 (1987), 277–334.Google Scholar
[13] Morales, M., Noetherian symbolic blow-up and examples in any dimension, preprint.Google Scholar
[14] Peskine, C. and Szpiro, L., Liaison des variétés algébriques, Invent. Math., 26 (1974), 271–302.Google Scholar
[15] Rao, A. P., On self-linked curves, Duke Math. J., 49 (1982), 251–273.Google Scholar
[16] Roberts, P., A prime ideal in a polynomial ring whose symbolic blow-up is not Noetherian, Proc. Amer. Math. Soc., 94 (1985), 589–592.Google Scholar
[17] Robbiano, L. and Valla, G., Free resolutions for special tangent cones, in “Commutative algebra, Proc. Trento Conf.”, eds.: Greco, S. and Valla, G., Lecture Notes in Pure and Appl. Math., Vol. 84, Dekker, New York, 1984, 253–274.Google Scholar
[18] Schenzel, P., Filtrations and Noetherian symbolic blow-up rings, to appear in Proc. Amer. Math. Soc. Google Scholar
[19] Schenzel, P., Examples of Noetherian symbolic blow-up rings, Rev. Roumaine Math. Pures Appl., 33 (1988), 4, 375–383.Google Scholar
[20] Szpiro, L., “Equations defining space curves”, Tata Institute of Fundamental Research, Bombay, Springer, Berlin, New York, 1979.Google Scholar
[21] Valla, G., On determinantal ideals which are set-theoretic complete intersections, Comp. Math., 42 (1981), 3–11.Google Scholar
[22] Valla, G., On set-theoretic complete intersections, in “Complete Intersections, Acireale 1983”, eds.: Greco, S. and Strano, R., Lecture Notes in Mathematics, Vol. 1092, Springer, Berlin, New York, 1984, 85–101.Google Scholar
[23] Vasconcelos, W., On linear complete intersections, J. Algebra, 111 (1987), 306–315.Google Scholar
[24] Vasconcelos, W., On the structure of certain ideal transforms, Math. Z., 198 (1988), 435–448.Google Scholar
[25] Vasconcelos, W., Symmetric algebras and factoriality, preprint.Google Scholar