Moduli Spaces of Marked Branched Projective Structures on Surfaces

TL;DR

This paper studies the moduli space of marked branched projective structures on surfaces, characterizing its dimension, singularities, and the structure of related branching class spaces, revealing their complex analytic and manifold properties.

Contribution

It introduces the notion of branching class, analyzes the structure of the moduli space, and describes its relation to branching classes, including dimension and singularity characterization.

Findings

Moduli space is a complex analytic space with dimension 6g - 6 + n for genus g surfaces.

Space of branching classes forms a complex manifold.

Relationships between the moduli space and branching classes depend on the branching degree n.

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.