On the associated graded ring of a semigroup ring

Let $(R,\mathbf m)$ be a numerical semigroup ring. In this paper we study the properties of its associated graded ring $G(\mathbf m)$. In particular, we describe the $H^0_{\mathcal M}$ for $G(\mathbf m)$ (where $\mathcal M$ is the homogeneous maximal ideal of $G(\mathbf m)$) and we characterize when $G(\mathbf m)$ is Buchsbaum. Furthermore, we find the length of $H^0_{\mathcal M}$ as a $G(\mathbf m)$-module, when $G(\mathbf m)$ is Buchsbaum. In the $3$-generated numerical semigroup case, we describe the $H^0_{\mathcal M}$ in term of the Apery set of the numerical semigroup associated to $R$. Finally, we improve two characterizations of the Cohen-Macaulayness and Gorensteinness of $G(\mathbf m)$ given in \cite{BF} and \cite{B}, respectively.