manuscripta mathematica

, Volume 61, Issue 3, pp 285–296 | Cite as

Gorenstein curves and symmetry of the semigroup of values

  • Félix Delgado de la Mata
Article

Abstract

LetO be the local ring of a irreducible algebroid curve and S its semigroup of values, Kunz in [7] proves thatO is a Gorenstein ring if and only if S is symmetrical. In this paper we give a generalization of this fact for the case of reduced curves with an arbitrary number of branches, d. For it we introduce a concept of symmetry for the semigroup of values S⊂ℤ+d which generalizes the well known symmetry for d=1 (i.e. the irreducible case). This concept of symmetry is also closely related to the symmetry introduced by García in [4] (for the d=2 case) and the author in [3] (for arbitrary d) with the main goal of the explicit determination of S (in the case of plane curves).

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    APERY, R.: Sur les branches superlineaires des courbes algébriques. C.R. Acad. Sci. Paris222, 1198–1200, (1945).Google Scholar
  2. [2]
    BASS, H.: On the ubiquity of Gorenstein rings. Math. Zeitschrit.82, 8–28, (1963).Google Scholar
  3. [3]
    DELGADO, F.: The semigroup of values of a curve singularity with several branches. Manuscripta Math.59, 347–374, (1987).Google Scholar
  4. [4]
    GARCIA, A.: Semigroups associated to singular points of plane curves. J. Reine. Angew. Math.336, 165–184, (1982).Google Scholar
  5. [5]
    GORENSTEIN, D.: An arithmetic theory of adjoint plane curves. Trans. Amer. Math. Soc.72, 414–436, (1952).Google Scholar
  6. [6]
    HERZOG, J.: Generators and relations of Abelian semigroups and semigroup rings. Manuscripta Math.3, (1970).Google Scholar
  7. [7]
    KUNZ, E.: The value-semigroup of a one dimensional Gorenstein ring. Proc. Amer. Math. Soc.25, 748–751, (1970).Google Scholar
  8. [8]
    SERRE, J.P.: Groupes algébriques et corps de classes. Hermann. Paris (1959).Google Scholar

Copyright information

© Springer-Verlag 1988

Authors and Affiliations

  • Félix Delgado de la Mata
    • 1
  1. 1.Dto. Algebra y GeometríaUniversidad de ValladolidValladolidSpain

Personalised recommendations