MathOverflow is a question and answer site for professional mathematicians. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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.

share|cite|improve this answer

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.