Nick Salter gave a great seminar here about this paper; hmm, maybe I should blog about that paper, which is really interesting, but I wanted to make a smaller point here. Let C be a smooth curve in P^2 of degree n. The lines in P^2 are parametrized by the dual P^2; let U be the open subscheme of the dual P^2 parametrizing those lines which are not tangent to C; in other words, U is the complement of the dual curve C*. For each point u of U, write L_u for the corresponding line in P^2.
This gives you a fibration X -> U where the fiber over a point u in U is L_u – (L_u intersect C). Since L_u isn’t tangent to C, this fiber is a line with n distinct points removed. So the fibration gives you an (outer) action of pi_1(U) on the fundamental group of the fiber preserving the puncture classes; in other words, we have a homomorphism
where B_n is the n-strand braid group.
When you restrict to a line L* in U (i.e. a pencil of lines through a point in the original P^2) you get a map from a free group to B_n; this is the braid monodromy of the curve C, as defined by Moishezon. But somehow it feels more canonical to consider the whole representation of pi_1(U). Here’s one place I see it: Proposition 2.4 of this survey by Libgober shows that if C is a rational nodal curve, then pi_1(U) maps isomorphically to B_n. (OK, C isn’t smooth, so I’d have to be slightly more careful about what I mean by U.)