  We study the Hurwitz action of the classical braid group on factorisations of
a Coxeter element c in a well-generated complex reflection group W. It is
well-known that the Hurwitz action is transitive on the set of reduced
decompositions of c in reflections. Our main result is a similar property for
the primitive factorisations of c, i.e. factorisations with only one factor
which is not a reflection. The motivation is the search for a geometric proof
of Chapoton's formula for the number of chains of given length in the
non-crossing partitions lattice NCP_W. Our proof uses the properties of the
Lyashko-Looijenga covering and the geometry of the discriminant of W.
