Algebraic & Geometric Topology

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

Barbu Berceanu, Daniela Anca Măcinic, Ştefan Papadima, and Clement Popescu

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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 Orlik–Solomon models associated to these quasiprojective manifolds to compute pairs of analytic germs at the origin, both for rank-1 and rank-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.

Article information

Algebr. Geom. Topol., Volume 17, Number 2 (2017), 1163-1188.

Received: 22 April 2016
Accepted: 7 July 2016
First available in Project Euclid: 19 October 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 55N25: Homology with local coefficients, equivariant cohomology 55R80: Discriminantal varieties, configuration spaces
Secondary: 14F35: Homotopy theory; fundamental groups [See also 14H30] 20F38: Other groups related to topology or analysis

partial configuration space smooth projective curve Gysin model admissible maps onto curves representation variety cohomology jump loci Malcev completion


Berceanu, Barbu; Măcinic, Daniela Anca; Papadima, Ştefan; Popescu, Clement. On the geometry and topology of partial configuration spaces of Riemann surfaces. Algebr. Geom. Topol. 17 (2017), no. 2, 1163--1188. doi:10.2140/agt.2017.17.1163.

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