Gorenstein curves and symmetry of the semigroup of values
- 86 Downloads
LetO be the local ring of a irreducible algebroid curve and S its semigroup of values, Kunz in  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  (for the d=2 case) and the author in  (for arbitrary d) with the main goal of the explicit determination of S (in the case of plane curves).
Unable to display preview. Download preview PDF.
- APERY, R.: Sur les branches superlineaires des courbes algébriques. C.R. Acad. Sci. Paris222, 1198–1200, (1945).Google Scholar
- BASS, H.: On the ubiquity of Gorenstein rings. Math. Zeitschrit.82, 8–28, (1963).Google Scholar
- DELGADO, F.: The semigroup of values of a curve singularity with several branches. Manuscripta Math.59, 347–374, (1987).Google Scholar
- GARCIA, A.: Semigroups associated to singular points of plane curves. J. Reine. Angew. Math.336, 165–184, (1982).Google Scholar
- GORENSTEIN, D.: An arithmetic theory of adjoint plane curves. Trans. Amer. Math. Soc.72, 414–436, (1952).Google Scholar
- HERZOG, J.: Generators and relations of Abelian semigroups and semigroup rings. Manuscripta Math.3, (1970).Google Scholar
- KUNZ, E.: The value-semigroup of a one dimensional Gorenstein ring. Proc. Amer. Math. Soc.25, 748–751, (1970).Google Scholar
- SERRE, J.P.: Groupes algébriques et corps de classes. Hermann. Paris (1959).Google Scholar