In this paper, we explore when the Betti numbers of the coordinate rings of a projective monomial curve and one of its affine charts are identical. Given an infinite field $k$ and a sequence of relatively prime integers $a_0 = 0 < a_1 < \cdots < a_n = d$, we consider the projective monomial curve $\mathcal{ C}\subset\mathbb{ P}_k^{ \,n}$ of degree $d$ parametrically defined by $x_i = u^{ a_i}v^{ d-a_i}$ for all $i \in \{ 0,\ldots,n\}$ and its coordinate ring $k[\mathcal{ C}]$. The curve $\mathcal{ C}_1 \subset \mathbb A_k^n$ with parametric equations $x_i = t^{ a_i}$ for $i \in \{ 1,\ldots,n\}$ is an affine chart of $\mathcal{ C}$ and we denote by $k[\mathcal{ C}_1]$ its coordinate ring. The main contribution of this paper is the introduction of a novel (Gr\"obner-free) combinatorial criterion that provides a sufficient condition for the equality of the Betti numbers of $k[\mathcal{ C}]$ and $k[\mathcal{ C}_1]$. Leveraging this criterion, we identify infinite families of projective curves satisfying this property. Also, we use our results to study the so-called shifted family of monomial curves, i.e., the family of curves associated to the sequences $j+a_1 < \cdots < j+a_n$ for different values of $j \in \mathbb N$. In this context, Vu proved that for large enough values of $j$, one has an equality between the Betti numbers of the corresponding affine and projective curves. Using our results, we improve Vu's upper bound for the least value of $j$ such that this occurs.
Categories: math.AC