Branched Projective Structures and Bundles of Projective Frames on Surfaces

We show that the description of the holomorphic $\mathbb{C}{\mathrm{P}}^1$-bundle associated to a holomorphic projective structure on a Riemann surface in terms of the principal bundle of projective $2$-frames extends very well to the setting of branched projective structures. This generalization reveals a space of parameters, each of which is associated to a branching class. The space of branched projective structures with a given branching class appears as a space of connections on a given $\mathbb{C}{\mathrm{P}}^1$-bundle, and is consequently an affine space. Finally, we study the map which to a branching class associates the corresponding $\mathbb{C}{\mathrm{P}}^1$-bundle with section.