Moduli spaces of marked branched projective structures on surfaces (Q6636577)

Let \(\mathcal P_g(n)\) be the moduli spaces of marked branched projective structures of genus \(g\) and branching degree \(n\). The author proves that \(\mathcal P_g(n)\) is a complex analytic space of dimension \(6g-6+n\) and if \(g\geq 2\) its singular points are characterized in terms of their monodromy.\N\NThe author also introduces the notions of branching class and shows that the moduli space \(\mathcal A_g(n)\) of marked branching classes of genus \(g\) and branching degree \(n\) is a complex manifold. Furthermore, the author proves that \(\mathcal P_g(n)\) is an affine bundle over \(\mathcal A_g(n)\) if \(n<2g-2\) and that \(\mathcal P_g(n)\) is an analytic subspace of \(\mathcal A_g(n)\) if \(n>4g-4.\)