Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 23G (Riemann Surfaces)

A rational function is a quotient $r=P/Q$ of polynomials with $Q\ne 0$. Cancel common factors. It determines a map on the Riemann sphere by assigning infinity at poles and assigning at infinity the limit in the sphere. Equivalently homogenize $P,Q$ to the same degree $d=\max (\mathrm{deg}P,\mathrm{deg}Q)$ and set $f([Z:W])=[P_h(Z,W):Q_h(Z,W)]$. They have no common projective zero: finite ones would contradict coprimality, and at infinity at least one has a nonzero leading coefficient. This also proves the map is holomorphic.

The degree of the nonconstant rational function is $d$. The degree of a holomorphic map of compact Riemann surfaces is the number of preimages of any target point, counted with local mapping multiplicity. For target $[a:b]$, the homogeneous equation $b{P}_h-a{Q}_h=0$ is a nonzero homogeneous polynomial of degree $d$. Over $\mathbb{C}$ it factors into $d$ linear factors, with multiplicity, giving exactly $d$ projective zeros. Locally its zero order is precisely the map's multiplicity, since the other target-coordinate denominator is nonzero. The two degrees are equal.

There is a Möbius transformation sending any ordered triple of distinct points to $0,1,\infty$. When all are finite one can use $T_A(z)=(z-a_1)(a_2-a_3)/[(z-a_3)(a_2-a_1)]$; the formulas for infinity are its limits. Define $T_B$ in the same way. Then $T_B^{-1}\circ T_A$ sends $A$ to $B$ and restricts to a biholomorphism $X\setminus A\to X\setminus B$.

A singular point would satisfy $p=0$, $p_w=2w=0$ and $p_z=-3z^2+2z+1=0$. Thus $w=0$ and $z=1$ or $-1/3$. But $p(1,0)=1$ and $p(-1/3,0)=-5/27$, both nonzero. The curve is smooth everywhere.

The complex implicit function theorem says that near a zero of a holomorphic function with a nonzero partial derivative, the corresponding variable is a uniquely defined holomorphic function of the others. Where $p_w\ne 0$, use $z$ as a local coordinate with $w=w(z)$; where $p_z\ne 0$, use $w$ with $z=z(w)$. On overlaps the coordinate transitions are holomorphic with nonzero derivatives. These charts give $Z$ a Riemann surface structure, and both coordinate projections are holomorphic in them.

A ramification point of a nonconstant holomorphic map is a point where in local coordinates the map has the form $u\mapsto u^e$ with $e\ge 2$; its ramification, or branching, order is $e-1$. For $g=w$, take $(z,w)=(1,i)$. Here $p_w=2i\ne 0$, so $z$ is a local coordinate. Differentiating $w(z{)}^2=z^3-z^2-z$ gives $w^{\prime }(1)=0$ and $2i{w}^{\prime \prime }(1)=4$. ThereforeThe nonzero quadratic coefficient gives local degree $\boxed{2}$ and branching order $\boxed{1}$. If “branching order” denotes local degree rather than its excess in a different convention, the corresponding value is two.