arXiv subjects

Gustave Billon

Publications and source records attributed to Gustave Billon.

2 recordsLinked to original sources

arXiv2023

Moduli Spaces of Marked Branched Projective Structures on Surfaces

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.

math.AG↗

arXiv2023

Branched Projective Structures and Bundles of Projective Frames on Surfaces

We show that the description of the holomorphic $\mathbb C \mathrm P^1$-bundle associated to a holomorphic projective structure on a Riemann surface in terms of the principal bundle of projective $2$-frames extends very well to the setting of branched projective structures. This generalization reveals a space of parameters, each of which is associated to a branching class. The space of branched projective structures with a given branching class appears as a space of connections on a given $\mathbb C \mathrm P^1$-bundle, and is consequently an affine space. Finally, we study the map which to a branching class associates the corresponding $\mathbb C \mathrm P^1$-bundle with section.

math.DG↗