Extension of a rational map from a smooth projective curve

Every rational map from a smooth projective curve to a projective variety extends uniquely to a morphism. At a missing point, the discrete valuation ring of the curve lets one divide homogeneous coordinates by their smallest valuation, leaving regular coordinates of which at least one is a unit.

Let $\nu :\tilde{C}\to C$ be the normalization of a nodal curve. Its rational inverse $C⇢\tilde{C}$ cannot extend over the node: the two branches give two distinct points of $\tilde{C}$, whereas a morphism can assign only one image to the node.