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).