The following is a well-known fact:

Theorem.The fundamental group of the complement of a non-singular curve of degree $d$ in the complex projective plane is cyclic of order $d$.

This was further generalised by Fulton, who proved that the fundamental group of a nodal curve is Abelian. (As far as I understand, Fulton used the result for non-singular curves in his proof.) I think that Zariski's idea of proof of Fulton's theorem works to prove the non-singular case, since every two non-singular curves are isotopic. (I gather this from Deligne's *Séminaire Bourbaki* on Fulton's theorem, but feel free to correct me!)

Another proof is using the braid monodromy technique: you prove that if a degree-$d$ curve has an order-$d$ tangent line, then the fundamental group of its complement is Abelian. (I learnt this from Cogolludo's notes on braid monodromy.) It could be that Zariski used this to prove his statement, so maybe both proofs I am aware of use braid monodromy in the end.

Here's a question I asked around a bit, googled, and thought about, but got no satisfactory answer:

Are there other (easier? more direct?) proofs of the theorem above?

If not, is there maybe a moral reason one should expect that this is hard?