Failure of rational-map extension on a singular curve
= Failure of rational-map extension on a singular curve
Let $\nu:\widetilde C\to C$ be the <normalization of a nodal curve>. Its rational inverse $C\dashrightarrow\widetilde C$ cannot extend over the node: the two branches give two distinct points of $\widetilde C$, whereas a morphism can assign only one image to the node.