Past exam of the mathematics course of the University of Cambridge / 2010 / ii / Paper 3 / 23G (Algebraic Geometry)

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.

Work geometrically over an algebraic closure; the degree and ramification do not change under this base extension. The coordinate squaring map lands in $W$ because its defining equation pulls back to that of $V$. Above a generic target point with all coordinates nonzero there are $2^3$ square-root choices, identified in pairs by simultaneous sign change. HenceThe effective sign-change group is $C_2\times C_2$. At a point with all coordinates nonzero its stabilizer is trivial, and the squaring map is locally étale, since characteristic is not two. Thus $e_p=1$ there. If one coordinate is zero, a sign change of that coordinate fixes the point, giving stabilizer order two. More explicitly, if $X_0=0$, use $t=X_0/X_2$ as a local parameter; the curve equation has nonzero derivative in the other coordinate, so this is valid, and the corresponding target parameter is $t^2$. ThereforeTwo coordinates cannot vanish on either curve. For each vanishing coordinate, the other-coordinate ratio solves $z^8=-1$, giving eight distinct points; there are 24 ramification points. As a check, the smooth plane-curve genera are $21$ and $3$, and Riemann-Hurwitz formula gives $2\cdot 21-2=4(2\cdot 3-2)+24$.