Suppose that $C\subset\mathbb P^2$ is an irreducible projective curve over $\mathbb C$ such that the normalization morphism $\bar C\to C$ is unramified (i.e., the induced morphism $\bar C\to\mathbb P^2$ is an immersion). Could you help me with examples of curves with this property for which $\pi_1(\mathbb P^2-C)$ is not commutative?

Thank you in advance.

unramifiednormalisation, because "homeomorphisms cannot be ramified"). Am I right that you are asking for an example of immersed plane curve $C: \,q(x, y, z)$ such that its Milnor fiber $F:\, q(x, y, z) = 1$ has nontrivial $\pi_1$? For irreducible curve $\pi_1(F)$ should be equal exactly to $\pi_1(P^2 \setminus C)'$, unless I'm making some obvious mistake again. (Numerous articles by Degtyarev, Shimada and Dimcashouldcontain multifarious examples..) $\endgroup$1more comment