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

Abstract

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 projective curves. Twisted Alexander polynomials with respect to finite $SU(2)$ representations show to be useful for this purpose, since all their abelian invariants are the same for both fundamental groups.