Minimal Graded Free Resolutions for Monomial Curves Defined by Arithmetic Sequences

TL;DR

This paper explicitly constructs minimal graded free resolutions for monomial curves defined by arithmetic sequences, revealing that their Betti numbers depend only on the initial term modulo the sequence length, confirming a conjecture on periodicity.

Contribution

It provides an explicit minimal free resolution for these monomial curves and proves the Betti numbers' periodicity depending solely on the initial term modulo the sequence length.

Findings

Betti numbers depend only on m_0 modulo n

Constructed explicit minimal graded free resolution

Confirmed Herzog and Srinivasan's periodicity conjecture

Abstract

Let $\mm=(m_0,...,m_n)$ be an arithmetic sequence, i.e., a sequence of integers $m_0<...<m_n$ with no common factor that minimally generate the numerical semigroup $\sum_{i=0}^{n}m_i\N$ and such that $m_i-m_{i-1}=m_{i+1}-m_i$ for all $i\in\{1,...,n-1\}$. The homogeneous coordinate ring $\Gamma_\mm$ of the affine monomial curve parametrically defined by $X_0=t^{m_0},...,X_n=t^{m_n}$ is a graded $R$-module where $R$ is the polynomial ring $k[X_0,...,X_n]$ with the grading obtained by setting $\deg{X_i}:=m_i$. In this paper, we construct an explicit minimal graded free resolution for $\Gamma_\mm$ and show that its Betti numbers depend only on the value of $m_0$ modulo $n$. As a consequence, we prove a conjecture of Herzog and Srinivasan on the eventual periodicity of the Betti numbers of semigroup rings under translation for the monomial curves defined by an arithmetic sequence.