Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 23G / i / Solution

At a smooth point of an irreducible curve, the local ring ${\mathcal{O}}_{X,p}$ is a discrete valuation ring. A local parameter is an element $t$ generating its maximal ideal; its order at $p$ is one. Embed the projective target as a closed subvariety of ${\mathbb{P}}^N$. Write the rational map generically as $[f_0:\cdots :f_N]$ with $f_i\in K(X)$, and let $m$ be the smallest finite valuation of the nonzero coordinates at $p$. Multiplying by $t^{-m}$ makes every coordinate regular at $p$ and at least one a unit. These functions are regular on some common neighborhood of $p$ and have no common zero after shrinking, so define a morphism to ${\mathbb{P}}^N$. The defining homogeneous equations of the target vanish generically and hence identically on this neighborhood. Thus the rational map extends to the projective target at $p$.

For a nonprojective target, $x\mapsto 1/x$ from ${\mathbb{A}}^1$ rationally to ${\mathbb{A}}^1$ cannot extend at zero: a regular extension would give $xf(x)=1$ there, impossible. In dimension two the morphism $(x,y)\mapsto [x:y]$ on ${\mathbb{C}}^2\setminus \{0\}$ cannot extend to zero, because its constant values along the two punctured coordinate axes are different. This shows why both projectivity of the target and smooth one-dimensional source were used.