Geodesic
On the geometry and topology of partial configuration spaces of Riemann surfaces
Berceanu, Barbu1 ; Măcinic, Daniela Anca1 ; Papadima, Ştefan1 ; Popescu, Clement1
1Institute of Mathematics “Simion Stoilow”, Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania
Algebraic and Geometric Topology, Tome 17 (2017) no. 2, pp. 1163-1188
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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.

DOI : 10.2140/agt.2017.17.1163
Classification : 55N25, 55R80, 14F35, 20F38
Keywords: partial configuration space, smooth projective curve, Gysin model, admissible maps onto curves, representation variety, cohomology jump loci, Malcev completion

Berceanu, Barbu  1   ; Măcinic, Daniela Anca  1   ; Papadima, Ştefan  1   ; Popescu, Clement  1

1 Institute of Mathematics “Simion Stoilow”, Romanian Academy, P.O. Box 1-764, 014700 Bucharest, Romania 
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Berceanu, Barbu; Măcinic, Daniela Anca; Papadima, Ştefan; Popescu, Clement. On the geometry and topology of partial configuration spaces of Riemann surfaces. Algebraic and Geometric Topology, Tome 17 (2017) no. 2, pp. 1163-1188. doi: 10.2140/agt.2017.17.1163

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