Cohen-Macaulayness of associated graded rings of Gorenstein monomial curves

Abstract

Let C be a Gorenstein noncomplete intersection monomial curve in the 4-dimensional affine space with defining ideal I(C). In this article, we use the minimal generating set of I(C) to give a criterion for determining whether the tangent cone of C is Cohen-Macaulay. We also show that if the tangent cone of C is Cohen-Macaulay, then the minimal number of generators of the ideal I(C)⁎ is either 5 or an even integer of the form 2d+2, for a suitable integer d. Additionally, we provide a family of Gorenstein noncomplete intersection monomial curves C whose tangent cone is Cohen-Macaulay and the minimal number of generators of I(C)⁎ is large.