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 $I(C)_{\ast}$ 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 $I(C)_{\ast}$ is large.