A note on Gorenstein monomial curves

Abstract

Let k be an arbitrary field. In this note, we show that if a sequence of relatively prime positive integers a = (a ₁, a ₂, a ₃, a ₄) defines a Gorenstein non complete intersection monomial curve $\mathcal{C}(a)$ in $\mathbb{A}_k^4$, then there exist two vectors u and v such that $\mathcal{C}(a + tu)$ and $\mathcal{C}(a + tv)$ are also Gorenstein non complete intersection affine monomial curves for almost all t ≥ 0.