Hostname: page-component-cd9895bd7-gbm5v Total loading time: 0 Render date: 2024-12-24T12:40:04.586Z Has data issue: false hasContentIssue false
  • English
  • Français

Seminormal rings and weakly normal varieties

Published online by Cambridge University Press:  22 January 2016

John V. Leahy  and
Marie A. Vitulli
John V. Leahy
Affiliation:
Institute of Theoretical Sciences and Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Marie A. Vitulli
Affiliation:
Department of Mathematics, University of Oregon, Eugene, Oregon 97403
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

In the late sixties and early seventies the operation of weak normalization was formally introduced first in the case of analytic spaces and later in the abstract scheme setting (cf. [6] & [4]). The notion arose from a classification problem. An unfortunate phenomenon in this area occurs when one tries to parametrize algebraic objects associated with a space by an algebraic variety; the resulting variety is, in general, not uniquely determined and may, for example, depend on the choice of coordinates. Under certain conditions one does know that the normalization of the parameter variety is unique. The price one pays for passing to the normalization is that often this variety no longer parametrizes what it was intended to; one point on the original parameter variety may split into several in the normalization. This problem is avoided if one passes instead to the weak normalization of the original variety. One then obtains a variety homeomorphic to the original variety with universal mapping properties that guarantee uniqueness.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 82 , June 1981 , pp. 27 - 56
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1981

References

[1] Adkins, W., Andreotti, A. and Leahy, J. V., An analogue of Oka’s theorem for weakly normal complex spaces, Pacific J. Math., 68 (1977), 297–301.Google Scholar
[2] Adkins, W., Andreotti, A. and Leahy, J. V., Weakly normal complex spaces (to appear).Google Scholar
[3] Adkins, W. and Leahy, J. V., A topological criterion for local optimality of weakly normal complex spaces (to appear).CrossRefGoogle Scholar
[4] Andreotti, A. and Bombieri, E., Sugli omeomorfismi delle varietà algebriche, Ann. Scuola Norm. Sup. Pisa, (1969), 430–450.Google Scholar
[5] Andreotti, A. and Holm, P., Quasi analytic and parametric spaces, Real and complex singularities, Oslo 1976, Sijthoff and Noordhoff Inter. Pub., Amsterdam (1977), 13–97.Google Scholar
[6] Andreotti, A. and Norguet, F., La convexité holomorphe dans l’espace analytique des cycles d’une variété algébrique, Ann. Scuola Norm. Sup. Pisa, 21 (1967), 31–82.Google Scholar
[7] Bombieri, E., Seminormalità a singolarità ordinaire, Symposia Mathematica, Academic Press (1973), 205–210.Google Scholar
[8] Endô, S., Projective modules over polynomial rings, J. Math. Soc. Japan, 15 (1963), 339–352.Google Scholar
[9] Gilmer, R. and Heitmann, R. C., On Pic(R[X]) for R seminormal (to appear).CrossRefGoogle Scholar
[10] Hamann, E., On the R-invariance of R[X] , J. Alg., 35 (1975), 1–17.Google Scholar
[11] Hartshorne, R., Algebraic Geometry, Springer GTM 52, New York (1977).Google Scholar
[12] Lipman, J., Relative Lipschitz-saturation, Am. J. Math., 97 (1975), 791–813.Google Scholar
[13] Matsumura, H., Commutative Algebra, W. A. Benjamin, New York (1970).Google Scholar
[14] Mochizuki, N., Holomorphic extension of continuous, weakly holomorphic functions on certain analytic varieties, Nagoya Math. J., 52 (1973), 53–59.CrossRefGoogle Scholar
[15] Nagata, M., Local Rings, Interscience Publishers (John Wiley), New York (1962).Google Scholar
[16] Orecchia, F., Sulla seminormalità di certe varietà affini riducibli, Boll. U.M.I. (5) 13-B (1976), 588–600.Google Scholar
[17] Palais, R., Some analogues of Hartogs’ theorem in an algebraic setting, Am. J. Math., 100 (1978), 387–406.CrossRefGoogle Scholar
[18] Serre, J. P., Géométrie Algébrique et Géométrie Analytique, Ann. Inst. Fourier 6 (1956), 1–42.Google Scholar
[19] Shafarevich, I. R., Basic Algebraic Geometry, Springer, New York (1974).Google Scholar
[20] Traverso, C., Seminormality and Picard group, Ann. Scuola Norm. Sup. Pisa, 24 (1970), 585–595.Google Scholar