Suppose that $F$ is a non-singular homogeneous polynomial in $\mathbb{C}^3$ and let $X$ be its zero locus, which is well-defined in $\mathbb{P}^2$. Consider the function $\pi:X\rightarrow\mathbb{P}^1$ that sends $[x:y:z]\in X$ to $[x:y]$.

My question is: how can I compute the multiplicity of $\pi$ on its branch points? I know that these branch points happen in the points $p\in X$ such that $\dfrac{\partial F}{\partial z}(p)=0$.

My first approach was that those points would need to satisfy $F(x,y,z)=0$ and $\dfrac{\partial F}{\partial z}(x,y,z)=0$ (note that $\dfrac{\partial F}{\partial z}$ is homogeneous of degree $d-1$), and therefore they would be the intersection points of two zero loci... but then I found out that I am messing up the terms "multiplicity" here.

I cannot use the divisor theory to show this result, but to me it seems right to have that the multiplicity of every branch point is $d$; for this it would be sufficient to show that there's only one pre-image for every branch point (and since the degree of $\pi$ is $d$, $d$ would be its multiplicity).

I would welcome any help in this matter; one can see that, from my last three questions, that I'm not very proficient with this subject, so if explanations were quite clear I'd be very satisfied :)


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