Buchsbaumness, Macaulayfication and Castelnuovo-Mumford regularity of monomial curves

TL;DR

This paper constructs an infinite class of non-smooth, non Cohen-Macaulay projective monomial curves that are k-Buchsbaum, finds their Macaulayfication generators, and explores their Castelnuovo-Mumford regularity in relation to k-Buchsbaumness.

Contribution

It introduces a method to determine the Macaulayfication of k-Buchsbaum monomial curves for any k and analyzes their Castelnuovo-Mumford regularity.

Findings

Constructed infinite class of non-smooth, non Cohen-Macaulay k-Buchsbaum curves.

Provided explicit monomial generators for their Macaulayfication.

Linked Castelnuovo-Mumford regularity to k-Buchsbaumness.

Abstract

Projective monomial curves are associated with rings generated by monomials of equal degree in two variables. In this paper, we give an infinite class of non-smooth, non Cohen-Macaulay $k$-Buchsbaum projective monomial curves for any $k\geq 1$ and find the monomial generators for the respective Macaulayfication. More generally, we demonstrate a method to find the Macaulayfication of a $k$-Buchsbaum monomial curve for any $k\geq 1$. We also discuss Castelnuovo-Mumford regularity of certain curves in terms of $k$-Buchsbaumness.