Past exam of the mathematics course of the University of Cambridge / 2006 / ii / Paper 3 / 22F / Solution

For a nonconstant holomorphic map between compact Riemann surfaces, choose local coordinates making the map $z\mapsto z^{e_p}$. Its local degree is $e_p$, and its ramification excess is $e_p-1$; branching order conventions use either of these quantities, so both are specified below. The degree $d$ is the number of inverse images of any regular value, or the sum of local degrees in any fiber. The Riemann-Hurwitz formula is $2g_X-2=d(2g_Y-2)+{\sum }_p(e_p-1)$.

The projective closure is $S^m-T^m-Z^m=0$. Its three partial derivatives cannot vanish simultaneously at any projective point, so it is nonsingular and inherits the complex structure of a smooth projective curve. The affine polynomial is irreducible by Eisenstein applied to $t^m-(s^m-1)$ at the prime $s-1$ in $\mathbb{C}[s]$, so the smooth projective curve is connected. At $Z=0$ its points are $[\zeta :1:0]$, where ${\zeta }^m=1$; these are exactly the $m$ points added at infinity.

The extension of the coordinate projection is $F=[S:Z]$. There is no point on the curve with $S=Z=0$, so this gives a holomorphic map everywhere. More explicitly, in the chart $T=1$ put $\sigma =S/T$, $\tau =Z/T$. Near an infinity point, ${\sigma }^m=1+{\tau }^m$ and $\sigma (0)=\zeta \ne 0$. Thus $\tau$ is a local coordinate and $1/F=\tau /\sigma$ has nonzero derivative $1/\zeta$ at zero. Each infinity point is unramified, with local degree one.

For a generic finite $s$, there are $m$ distinct solutions of $t^m=s^m-1$, so $\mathrm{deg}F=m$. Ramification occurs exactly at $t=0$, $s^m=1$. At each such point, $t$ is a local coordinate and $s-s_0=t^m/(m{s}_0^{m-1})+O(t^{2m})$, giving local degree $m$ and ramification excess $m-1$. There are no other critical points, since away from $t=0$ the projection $s$ is a local coordinate. Therefore