# Branch Points of Riemann Surfaces

Can a Riemann surface of a complex-valued function have three branch points? I've been learning about Riemann surfaces from Brown's complex analysis book and the exposition isn't too general, so if the answer is yes I'd appreciate not just an example but some of the intuition behind how many branch points a given Riemann surface can have.

## 1 Answer

Consider the algebraic curve $X$ in $\mathbb{C}^2$ defined by the zeroes of the polynomial $p(z,w)=w^3-z(z^2-1)$. This can be made into a Riemann surface as a consequence of the Implicit Function Theorem, as you probably know. Now define $f:X\rightarrow\mathbb{C}$ by $f(z,w)=z$. Then $f$ has degree 3. However the points $z=0,\pm 1$ have only a single preimage in $X$. Hence they are branch points with branching order 3.

In terms of more general theory I think it makes more sense once you've done some algebraic geometry (which I don't know that much about yet, sadly)! However, heuristically it does seem appropriate that projection maps from algebraic curves defined by cubics should have 3 branch points. More generally you can see how to construct maps with $n$ branch points.