Abstract:
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 Betti numbers of $k[\mathcal{C}]$ and $k[\mathcal{C}_1]$ satisfy the relation $\beta_i(k[\mathcal{C}_1]) \leq \beta_i(k[\mathcal{C}])$ for all $i$. In this talk, we will discuss when the Betti numbers of the coordinate rings of a projective monomial curve and one of its affine charts are identical. This talk is based on a joint work with Ignacio García-Marco and Philippe Gimenez.