Projective closure of Gorenstein monomial curves and the Cohen-Macaulay property

Let C( a) be a Gorenstein non-complete intersection monomial curve in the 4-dimensional affine space. There is a vector v ∈ N⁴ such that for every integer m ≥ 0, the monomial curve C( a+m v) is Gorenstein non-complete intersection whenever the entries of a+m v are relatively prime. In this paper, we study the arithmetically Cohen-Macaulay property of the projective closure of C( a+m v).