On the geometry and topology of partial configuration spaces of Riemann surfaces

TL;DR

This paper investigates the geometric and topological properties of partial configuration spaces of Riemann surfaces, focusing on their fundamental groups, representation varieties, and associated algebraic structures.

Contribution

It introduces Gysin models to compute local analytic invariants and describes regular maps onto curves, revealing the role of curves of general type in the structure.

Findings

Explicit descriptions of representation varieties at the origin

Classification of regular surjections onto curves of negative Euler characteristic

Finite presentations for Malcev Lie algebras of fundamental groups

Abstract

We examine complements (inside products of a smooth projective complex curve of arbitrary genus) of unions of diagonals indexed by the edges of an arbitrary simple graph. We use Gysin models associated to these quasi-projective manifolds to compute pairs of analytic germs at the origin, both for rank 1 and 2 representation varieties of their fundamental groups, and for degree 1 topological Green--Lazarsfeld loci. As a corollary, we describe all regular surjections with connected generic fiber, defined on the above complements onto smooth complex curves of negative Euler characteristic. We show that the nontrivial part at the origin, for both rank 2 representation varieties and their degree 1 jump loci, comes from curves of general type, via the above regular maps. We compute explicit finite presentations for the Malcev Lie algebras of the fundamental groups, and we analyze their formality properties.