Complements to duals of smooth curves

Question

Suppose that $C\subset \mathbb P^2$ is a smooth curve (over $\mathbb C$) and $C^*\subset (\mathbb P^2)^*$ is its dual curve. What is known about $\pi_1$ of the complement to $C^*$ in $(\mathbb P^2)^*$? I am particularly interested in smooth curves of low degree.

Answer 1

One of the invariants of the fundamental group of the complement of a plane curve is the Alexander polynomial, and this invariant is easier to access than the fundamental group itself. This invariant is not too hard to calculate in concrete examples but is still not that well understood.

In the particular case where $C^*$ is a degree 6 curve with 9 ordinary cusps, i.e., the dual of a smooth cubic, th Alexander polynomial equals $(t^2-t+1)^3$, which implies that the fundamental group is highly non-abelian.