Algebraic invariants of projective monomial curves associated to generalized arithmetic sequences

Let $K$ be an infinite field and let $m_1,\dots ,m_n$ be a generalized arithmetic sequence of positive integers, i.e., there exist $h, d, m_1 \in$\mathbb{Z}$^+$suchthat$m_i = h m_1 + (i-1)d$forall$i \in {2,\ldots,n}$\mathrm{.}Weconsidertheprojectivemonomialcurve$\mathcal C\subset $\mathbb{P}$^{n}_{K}$parametricallydefinedby$$x_1=s^{m_1}{t}^{m_n-m_1},\dots ,x_{n-1}=s^{m_{n-1}}{t}^{m_n-m_{n-1}},x_n=s^{m_n},x_{n+1}=t^{m_n}\mathrm{.}$$ In this work, we characterize the Cohen-Macaulay and Koszul properties of the homogeneous coordinate ring $K[\mathcal{C}]$ of $\mathcal{C}$. Whenever $K[\mathcal{C}]$ is Cohen-Macaulay we also obtain a formula for its Cohen-Macaulay type. Moreover, when $h$ divides $d$, we obtain a minimal Gr"obner basis $\mathcal{G}$ of the vanishing ideal of $\mathcal{C}$ with respect to the degree reverse lexicographic order. From $\mathcal{G}$ we derive formulas for the Castelnuovo-Mumford regularity, the Hilbert series and the Hilbert function of $K[\mathcal{C}]$ in terms of the sequence.