I'm looking at the Riemann surface $R = \{(z,w) : f(z,w) = w^3 - z^3 + z = 0 \}$. I'm looking at the projection map $f: (z,w) \to z$ and I'm trying to find the degree and the branching index. I can see that whenever $w \ne 0$ we have local coordinates $z \to (z,g(z))$ and hence $f_{loc}(z) = z$ but when $w=0$ we have local coordinates $w \to (g(w), w)$ for some holomorphic map $g$ and then by the chain rule on $f(g(w),w)$ we see that $g'(w) = 0$ and so $(0,0),(-1,0),(1,0)$ are the ramification points.
Now to find the degree I'm looking at $\deg(f) = \sum v_{f}(r)$ where the sum is taken over all the $r$ in some pre-image $f^{-1}(s)$ for fixed s. So let's say I take $s=0$ and then there are three points in the pre-image as listed above and they all have valency 3 and hence the degree is 9. But I have in my notes that the answer should be 3 but I cannot see how this is the case?
Thanks for any help with this