Moduli spaces of marked branched projective structures on surfaces

We show that the moduli space 𝒫 g (n) of marked branched projective structures of genus g and branching degree n is a complex analytic space. In the case g≥2, we show that 𝒫 g (n) is of dimension 6g-6+n and we characterize its singular points in terms of their monodromy. We introduce a notion of branching class, that is an infinitesimal description of branched projective structures at the branched points. We show that the space 𝒜 g (n) of marked branching classes of genus g and branching degree n is a complex manifold. We show that if n<2g-2 the space 𝒫 g (n) is an affine bundle over 𝒜 g (n), while if n>4g-4, 𝒫 g (n) is an analytic subspace of 𝒜 g (n).