For any increasing sequence \(m_1 < \dots < m_n \subset \mathbb{N}\) of integers, one can define a curve in projective \(n\)-space parametrically by setting \[ x_i = s^{m_i} t^{m_n-m_i} \] for \(i=1,\dots,n\) and \(x_{n+1} = t^{m_n}\). The vanishing ideal of the curve is a binomial prime (i.e. toric) ideal and often it encodes features of the sequence of integers.
In this paper the authors study the case when the sequence is a generalized arithmetic sequence, that is, \(m_i = hm_1 + (i-1)d\) for some fixed integers \(h,d\). Among the main results are a reverse lexicographic Gröbner basis of the toric ideal and necessary and sufficient condition for the coordinate ring of the monomial curve to be Cohen-Macaulay (\(d\) must equal one according to Corollary 3.2) and Koszul (\(m_i = m_{i-1}+1\) for all \(i\) and \(n>m_1\) according to Theorem 4.1).