Projective closures of affine monomial curves

TL;DR

This paper investigates the projective closures of three key affine monomial curves in four dimensions to understand the relationship between their syzygies and the arithmetic Cohen-Macaulay property.

Contribution

It provides new insights into the connections between syzygies and Cohen-Macaulayness for specific families of affine monomial curves.

Findings

Analysis of the Backelin, Bresinsky, and Arslan curves' projective closures

Identification of conditions linking syzygies to Cohen-Macaulay property

Potential classification criteria for these curves based on their algebraic properties

Abstract

We study the projective closures of three important families of affine monomial curves in dimension $4$, namely the Backelin curve, the Bresinsky curve and the Arslan curve, in order to explore possible connections between syzygies and the arithmetic Cohen-Macaulay property.