Branched projective structures with Fuchsian holonomy

We prove that if S is a closed compact surface of genus g >= 2, and if rho: pi(1)(S) -> PSL(2, C) is a quasi-Fuchsian representation, then the space M-k,M-rho of branched projective structures on S with total branching order k and holonomy rho is connected, for k > 0. Equivalently, two branched projective structures with the same quasi-Fuchsian holonomy and the same number of branch points are related by a movement of branch points. In particular grafting annuli are obtained by moving branch points. In the appendix we give an explicit atlas for M-k,M-rho for non-elementary representations rho. It is shown to be a smooth complex manifold modeled on Hurwitz spaces.