Complete intersection monomial curves and the Cohen-Macaulayness of their tangent cones

Let $C(\mathbf{n})$ be a complete intersection monomial curve in the 4-dimensional affine space. In this paper we study the complete intersection property of the monomial curve $C(\mathbf{n}+w\mathbf{v})$, where $w>0$ is an integer and $\mathbf{v}\in {\mathbb{N}}^4$. Also we investigate the Cohen-Macaulayness of the tangent cone of $C(\mathbf{n}+w\mathbf{v})$.