For a sequence of coprime positive integers \(\mathbf n =(n_1,\dots,n_d)\in\mathbb N^d\), the author studies properties of the associated graded ring of the local ring \(K[[ t^{n_1},\dots,t^{n_d}]]\), which turns out to be the ring of the tangent cone at the origin \(\Gamma(\mathbf n)\) of the monomial curve \(C(\mathbf n)\) defined by parametric equations \(x_1=t^{n_1},\dots,x_d=t^{n_d}\).
In particular, the author starts with fixed \(d\)-uples \(\mathbf n,\mathbf v\) and looks for values \(w\in \mathbb N\) such that \(\Gamma(\mathbf n+w\mathbf v)\) is Cohen-Macaulay or complete intersection. The author finds several results for the case \(d=4\). For instance, the author constructs vectors \(\mathbf v\) such that \(\Gamma(\mathbf n+w\mathbf v)\) is complete intersection whenever the entries of \(\mathbf n+w\mathbf v\) are coprime and \(w\) is greater than a fixed value \(w_0\). The author also provides an infinite family of complete intersection monomial curves \(C_m(\mathbf n+w\mathbf v)\) whose corresponding local rings have non-decreasing Hilbert functions, although their tangent cones are not Cohen-Macaulay.