For $h$ an odd degree polynomial, $\{(z,w)\in\mathbb{C}^2\mid w^2=h(z)\}$ can be made into a compact Riemann surface by adding 1 point at inifinty I want to prove that for $h$ an odd degree polynomial, $S=\{(z,w)\in\mathbb{C}^2\mid w^2=h(z)\}$ can be made into a compact Riemann surface by adding 1 point at inifinty.
My problem is that I can visually see this by drawing the cuts and gluing two sheets etc, however, I want to do this rigorously, but have no idea how to do this. Any outline of how to formalize this would be much appreciated.
 A: First of all, I think you should assume that $h$ has no multiple roots, otherwise you get singularities at those points. Which tools are you allowed to use here? E.g., the fact that $S$ itself is a Riemann surface basically follows from the complex implicit function theorem. For the compactification it is enough to show that your surface has only one non-compact "end" which is conformally isomorphic to a punctured disk. In order to do this, you can introduce new coordinates $(s,t)$ by the transformation $s = 1/z$ and $t = w/z^{m+1}$, where $2m+1$ is the degree of $h$. This transformation is a biholomorphic self-map of $(\mathbb{C\setminus \{0\}})^2$ which maps neighborhoods of $z=\infty$ to neighborhoods of $s=0$. If $h(z) = \sum_{k=0}^{2m+1} h_k z^k$, then the equation $w^2 = h(z)$ becomes $t^2 = \sum_{k=0}^{2m+1} h_k s^{2m+2-k} = s(h_{2m+1} + h_{2m} s + \ldots)$, so near $s=0$ you can use $t$ as a local parameter and you get a simple puncture for $t=s=0$.
(By the way, these objects are called elliptic curves if the degree is $3$ or $4$, and hyperelliptic curves if the degree is $5$ or larger, and there is lots of literature about them in libraries and on the Internet. For even degree $n = 2m+2$ you can use the same transformation, but the transformed equation becomes $t^2 = h_{2m+2} + h_{2m+1} s + \ldots$, so near $s=0$ you get two different branches of the square root with two different punctures at $t=\pm \sqrt{h_{2m+2}}$, so you have to add two points at $\infty$ to compactify the surface.)
