Tangent Cone of Numerical Semigroup Rings of Embedding Dimension Three

In this paper, we give new characterizations of the Buchsbaum and Cohen-Macaulay properties of the tangent cone ${\text{gr}}_{\text{frakm}}(R)$, where $(R,\text{frakm})$ is a numerical semigroup ring of embedding dimension 3. In particular, we confirm the conjectures raised by Sapko on the Buchsbaumness of ${\text{gr}}_{\text{frakm}}(R)$.