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

TL;DR

This paper investigates the Cohen-Macaulay property of the projective closure of a family of Gorenstein monomial curves in four-dimensional space, focusing on how this property behaves under certain algebraic transformations.

Contribution

It introduces a method to analyze the Cohen-Macaulay property of projective closures of Gorenstein monomial curves after specific shifts, extending understanding of their algebraic and geometric properties.

Findings

The projective closure retains Cohen-Macaulay property under certain conditions.

A characterization of when the shifted curves are arithmetically Cohen-Macaulay.

Insights into the structure of Gorenstein monomial curves in higher dimensions.

Abstract

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