Moduli Spaces of Marked Branched Projective Structures on Surfaces
Abstract
We show that the moduli space of marked branched projective structures of genus g and branching degree n is a complex analytic space. In the case g > 1 we show that this moduli space is of dimension 6 g - 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 of marked branching classes of genus g and branching degree n is a complex manifold. We show that if n < 2g-2 the space of branched projective structures is an affine bundle over the space of branching classes, while if n > 4g-4 the former is an analytic subspace of the latter.
Cite
@article{arxiv.2206.01419,
title = {Moduli Spaces of Marked Branched Projective Structures on Surfaces},
author = {Gustave Billon},
journal= {arXiv preprint arXiv:2206.01419},
year = {2023}
}
Comments
A mistake in the main theorem of the first version has been corrected : the moduli space of branched projective structures is not smooth, the singular points are identified. The case 2g-3 < n < 4g-3 has been added