Past exam of the geometry course of the University of Oxford / 2007 / B / 7

With local indices $e_p$, the required specialization isOnly the finitely many points of index greater than one contribute. Their images are the branch points on the target.

For $d>1$, the map $z\mapsto z^d$ has local index $d$ at zero and at infinity; at infinity use coordinates $w=1/z$ in source and target, giving again $w\mapsto w^d$. At every nonzero finite point its derivative is nonzero, so there are no further ramification points. Thus the ramification sum is $2(d-1)$ and the formula gives $-2=-2d+2(d-1)$, as required for the sphere. For $d=1$ the map is the identity and has no ramification, giving the same equality separately.

For a smooth complex projective plane curve of degree $d$,Project generically from a point outside the curve to a projective line. Its degree is $d$, by intersection with a generic line through the center. Ramification occurs when such a line is tangent. In coordinates for projection $[x:y:z]\mapsto [x:z]$, these points satisfy $F=F_y=0$. The polar has degree $d-1$, and Bézout gives total ramification $d(d-1)$; generic projection makes the ramification simple, or the intersection multiplicities can be read as the corresponding ramification orders. The Riemann-Hurwitz formula yields $2g-2=-2d+d(d-1)$ and hence the formula. The smoothness qualification is essential for geometric genus: singular plane curves have the same arithmetic degree-genus expression, while the genus of their normalization can be smaller.

The ramification sum equals $2g-2+2d$. Since $g\ge 0$ and $d>1$, it is at least $2d-2>0$. Thus some point has analytic ramification index greater than one.

For a degree-one map, the sum of local degrees in every fiber is one. Therefore each fiber contains exactly one point and its local degree is one. There is no ramification, and the Riemann-Hurwitz formula gives $2g-2=-2$, so $g=0$. The map is a continuous bijection from compact $C$ onto the Hausdorff sphere and hence a homeomorphism. In fact its local degree-one coordinate inverses are holomorphic, so it is biholomorphic as well.

For degree two every ramification point has index two, so contributes exactly one. The formula gives $2g+2$ ramification points. No two can share an image: that would give a fiber of total local degree at least four. Hence there are also $2g+2$ distinct branch points on the sphere.