On the topology of fiber-type curves: a Zariski pair of affine nodal curves

In this paper we explore conditions for a curve in a smooth projective surface to have a free product of cyclic groups as the fundamental group of its complement. It is known that if the surface is $\mathbb{P}^2$, then such curves must be of fiber type, i.e. a finite union of fibers of an admissible map onto a complex curve. In this setting, we exhibit an infinite family of Zariski pairs of fiber-type curves, that is, pairs of plane projective fiber-type curves whose tubular neighborhoods are homeomorphic, but whose embeddings in $\mathbb{P}^2$ are not. This includes a Zariski pair of curves in $\mathbb{C}^2$ with only nodes as singularities (and the same singularities at infinity) whose complements have non-isomorphic fundamental groups, one of them being free. Our examples show that the position of nodes also affects the topology of the embedding of fiber-type curves.
Comment: 17 pages. Comments are welcome and greatly appreciated. V2: Improved exposition, added examples