A proof from Janos Kollar's Lectures on Resolution of Singularities Kollar (p 37) works as follows:


>**Theorem 1.58** (M. Noether, 1871). Let $k$ be an algebraically closed
field and $C \subset \mathbb{P}^2$ an irreducible plane curve. Then the algorithm (1.57)
eventually stops with a curve $C_m \subset \mathbb{P}^2$, which has only ordinary multiple
points (1.54).


>We give two proofs. The first one assumes that we already know embedded
resolution as in (1.52). The second, following Noether’s original
approach, gives another proof of embedded resolution.

>*Proof using resolution.* Pick any point $p \in C \subset \mathbb{P}^2$, and let $\pi : C \to \mathbb{P}^1$ be
the projection from $p$. In characteristic zero or if the characteristic does not divide $\operatorname{deg} C − \operatorname{mult}_p C$, the projection $\pi$ is separable.

(This fact we can look up on page 15 but also simply assumed as black box: i.e. from now we assume that $\pi$ is separable: field extension $K(C)/K(\overline{\pi(C)}$ is separable.)

Now the part I not understand:

>*Thus we can take two general lines through $p$ such that they are not contained in the tangent cone of $C$ at $p$ and they have only transverse intersections with $C$
at other points.* [...]

**Question:** Why separability of $\pi$ is neccessary to be able to choose these two lines with described properties and what fails if $\pi$ would be not separable?

We know that seprableness of $\pi$ implies that a general fiber of $\pi$ has $\operatorname{deg} \pi$ points. Possibly, if $\operatorname{deg} \pi >1$ this might be give rise for two "candidates" for the two lines trough $p$ with desired properties, but it can also happen that $\operatorname{deg} \pi =1$ & $\pi$ separable. Thus I unfortunately not see the crucialness of separateness of $\pi$ for the justification of the existence of the the lines.