Let $k$ be an algebraically closed field, char $k=0$, and let $C\subset\mathbb{P}_k^2$ be a nonsingular projective plane curve of degree $d$.
Let $O\in C$, $L\subset\mathbb{P}_k^2$ a line not containing $O$, and consider the map $\varphi:C\to L$ given by projection away from $O$ onto $L$. This is a priori only a map $C-\{O\}\to L$, and then we extend it using that $C$ is a curve. It is a map of degree $d-1$.
I want to check this claim: the ramification index of $\varphi$ at $O$ is the order of vanishing of the tangent line to $C$ at $O$ (call if $f$) minus 1.
For, suppose that $\varphi(O)=P$. Then $\varphi^{-1}(O)=\{Q_1,\dots,Q_{r},P\}$, and $Q_1,\dots,Q_r$ are points on $C\cap OP$, where $OP$ is line between $O$ and $P$, and there are $d$ points on this line, counting multiplicities. If $e_i$ is the ramification index of $\varphi$ at $Q_i$, and $e$ is the ramification index of $\varphi$ at $P$, then we have $e+\sum_{i=1}^re_i=d-1$. On the other hand, $e_i$ is equal to the order of vanishing of $OP$ at $Q_i$, and we have that $d=f+\sum_{i=1}^re_i$. So we get $e=f-1$.
Now that I've written this I'm pretty sure I believe it, but I'd appreciate anyone who wants to check it!
