Degree of the projection map of projective plane curve to projective line

Question

Let $\mathbb{P}^1$ be the complex projective line and $\mathbb{P}^2$ be the complex projective plane. Let $\mathcal{C} \subset \mathbb{P}^2$ be a compact projective plane curve, i.e. the zero locus of a homogeneous polynomial $F \in \mathbb{C}[X,Y,Z]$ of degree $d$. We consider the holomorphic map between Riemann surfaces $\pi : \mathcal{C} \to \mathbb{P}^1, \, [x:y:z] \mapsto [x:z]$. I want to show that the degree of this map is equal to $d$, as it is stated in "Algebraic Curves and Riemann Surfaces" by R. Miranda (Plücker's formula, page 144) and in this post (without further details) : Genus of a smooth projective curve.

My attempt is the following. We take $p = [x_0,z_0] \in \mathbb{P}^1$ such that it is not the image by $\pi$ of a ramification point (it is possible as ramifications points are a finite subset of $\mathcal{C}$, the latter being compact). Then the degree of $\pi$ is just the number of preimages of $p$. Without loss of generalities we suppose $x_0 \neq 0$ so we can rewrite $p = [1:w_0]$. Now a preimage of $p$ is of the form $[1:y:w_0]$ and such that $F(1,y,w_0) = 0$. If $f(Y) = F(1,Y,w_0) \in \mathbb{C}[Y]$ we see that preimages of $p$ are in bijection with roots of $f$.

So I'd like to show that $f$ has $d$ distinct roots, but I don't see how to prove this (and I'm not 100% sure it is the right statement). Any help is welcome !

Answer 1

I helps me a lot to think about this more geometrically: You can embed $\mathbb P^1$ as a line $L$ into $\mathbb P^2$ by $$\mathbb P^1 \ni [x:z] \mapsto [x:0:z] \in L \subset \mathbb P^2.$$ With that embedding in mind, you may think of your map $\pi: \mathbb P^2 \setminus \{[0:1:0]\} \to \mathbb P^1$ as the projection from the point $[0:1:0]$ to $L$. This means that the image point $\pi([x:y:z])\in L$ is the point obtained by intersecting the line $N$, which connects $[0:1:0]$ and $[x:y:z]$, with $L$. As $N$ is a line, it intersects $C$ in $d$ points, counted with multiplicitiy.

Here is the picture which I have in mind: