I am reading Andreas Gathmann's notes on plane algebraic curve. In particular, Proposition 5.16 on his notes says that a smooth curve of degree $d$ in $\mathbb{P}_{\mathbb{C}}^2$ has topological genus $d-1 \choose 2$.
The proof goes like this. Let $F$ be a smooth curve of degree $d$ in $\mathbb{P}_{\mathbb{C}}^2$. By a projective coordinate transformation, we can assume $[0:1:0] \notin V(F)$. Then define $\pi:V(F)\to \mathbb{P}_{\mathbb{C}}^1$ as $[x:y:z] \mapsto [x:z]$. So it looks like a covering space for $\mathbb{P}^1$ except for some points where $F$ and $\partial F/\partial y$ vanishes together.
Then comes the part I don't understand. It says that if we choose the coordinate transformation general enough, then we can ensure that at those places where $F$ and $\partial F/\partial y$ vanishes together, exactly two roots of $F(x,\cdot, z)$ will coincide. How can I see this is true?
Moreover, is this true if we are in $\mathbb{P}^n$ and I consider the projection $\pi:\mathbb{P}^n \to \mathbb{P}^{n-1}$, we can always find coordinate transformation to have exactly two roots of $F(z_1,\dots,z_{n-1},\cdot)$ at the place where $F$ and $\partial F/ \partial y$ vanishes together?