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

中文翻译：

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Gorenstein 单项曲线的投影闭合和 Cohen-Macaulay 性质

令\(C(\textbf{a})\)为4维仿射空间中的Gorenstein非完全相交单项式曲线。存在一个向量\(\textbf{v} \in {\mathbb {N}}^{4}\)使得对于每个整数\(m \ge 0\)，单项式曲线\(C(\textbf{只要\(\textbf{ a}+m\textbf{v}\)的条目互质，a}+m\textbf{v})\) 就是 Gorenstein 非完全交集。在本文中，我们研究了\(C(\textbf{a}+m\textbf{v})\)射影闭包的算术 Cohen-Macaulay 性质。