Minimal Graded Free Resolution for Monomial Curves in $\mathbb{A}^{4}$ defined by almost arithmetic sequences

TL;DR

This paper constructs a minimal graded free resolution for the coordinate ring of a monomial curve in four-dimensional affine space, defined by an almost arithmetic sequence, advancing understanding of its algebraic structure.

Contribution

It provides the first explicit minimal graded free resolution for monomial curves in $ ext{A}^4$ defined by almost arithmetic sequences.

Findings

Explicit minimal graded free resolution constructed

Enhanced understanding of algebraic structure of monomial curves

Applicable to numerical semigroups generated by almost arithmetic sequences

Abstract

Let $\mm=(m_0,m_1,m_2,n)$ be an almost arithmetic sequence, i.e., a sequence of positive integers with ${\rm gcd}(m_0,m_1,m_2,n) = 1$, such that $m_0<m_1<m_2$ form an arithmetic progression, $n$ is arbitrary and they minimally generate the numerical semigroup $\Gamma = m_0\N + m_1\N + m_2\N + n\N$. Let $k$ be a field. The homogeneous coordinate ring $k[\Gamma]$ of the affine monomial curve parametrically defined by $X_0=t^{m_0},X_{1}=t^{m_1},X_2=t^{m_3},Y=t^{n}$ is a graded $R$-module, where $R$ is the polynomial ring $k[X_0,X_1,X_3, Y]$ with the grading $\deg{X_i}:=m_i, \deg{Y}:=n$. In this paper, we construct a minimal graded free resolution for $k[\Gamma]$.