Given a sequence of relatively prime integers \(a_0 = 0 < a_1 < \dots < a_n = d\) and a field \(k\), 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 its coordinate ring by \(k[\mathcal{C}_1]\).

In this talk, we will discuss the equality of the Betti numbers of \(k[\mathcal{C}]\) and \(k[\mathcal{C}_1]\) and present a combinatorial criterion that ensures equality. This is joint work with I. García-Marco and P. Gimenez.