(2.1.2) Let $\mathscr{C}$ be a connected smooth algebraic curve over $k$. Let $s$ be a $k$-rational point on $\mathscr{C}$ and set $C=\mathscr{C}\setminus\{s\}$. We fix a uniformizer $t$ in $\mathcal{O}_{\mathscr{C},s}$. This choice determines an isomorphism of $k$-algebras $R\to\widehat{\mathcal{O}}_{\mathscr{C},s}$ and thus a morphism of $k$-schemes $\mathrm{Spec}\,R\to\mathscr{C}$.