Minimal Graded Free Resolution for Monomial Curves in ${\mathbb{A}}^4$ defined 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 $\gcd (m_0,m_1,m_2,n)=1$, such that $m_0<m_1<m_2$ form an arithmetic progression, $n$ is arbitrary and they minimally generate the numerical semigroup $\Gamma =m_0\mathbb{N}+m_1\mathbb{N}+m_2\mathbb{N}+n\mathbb{N}$. 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 $°X_i:=m_i,°Y:=n$. In this paper, we construct a minimal graded free resolution for $k[\Gamma ]$.

arXiv commentAccepted for publication in Communication in Algebra