On the Buchsbaumness of the associated graded ring of a one-dimensional local ring

Let (R, m) be a Noetherian, one-dimensional, local ring, with $|R/m|=\infty$. We study when its associated graded ring G(m) is Buchsbaum; in particular, we give a theoretical characterization for G(m) to be Buchsbaum not Cohen-Macaulay. Finally, we consider the particular case of R being the semigroup ring associated to a numerical semigroup S: we introduce some invariants of S, and we use them in order to give a necessary and a sufficient condition for G(m) to be Buchsbaum.