Hilbert functions of Gorenstein monomial curves


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Arslan F., Mete P.

PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY, vol.135, no.7, pp.1993-2002, 2007 (Journal Indexed in SCI) identifier identifier

  • Publication Type: Article / Article
  • Volume: 135 Issue: 7
  • Publication Date: 2007
  • Doi Number: 10.1090/s0002-9939-07-08793-x
  • Title of Journal : PROCEEDINGS OF THE AMERICAN MATHEMATICAL SOCIETY
  • Page Numbers: pp.1993-2002
  • Keywords: Gorenstein local ring, Hilbert function of a local ring, tangent cone, monomial curve, numerical semigroup, standard basis, ORDINARY SINGULARITIES, SEMIGROUPS, RING

Abstract

It is a conjecture due to M. E. Rossi that the Hilbert function of a one-dimensional Gorenstein local ring is non-decreasing. In this article, we show that the Hilbert function is non-decreasing for local Gorenstein rings with embedding dimension four associated to monomial curves, under some arithmetic assumptions on the generators of their de. ning ideals in the non-complete intersection case. In order to obtain this result, we determine the generators of their tangent cones explicitly by using standard basis computations under these arithmetic assumptions and show that the tangent cones are Cohen-Macaulay. In the complete intersection case, by characterizing certain families of complete intersection numerical semigroups, we give an inductive method to obtain large families of complete intersection local rings with arbitrary embedding dimension having non- decreasing Hilbert functions.