Past exam of the mathematics course of the University of Cambridge / 2009 / ii / Paper 1 / 23G / b / Solution

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.