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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 Orlik-Solomon 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.

The complete classification of (3, 3)-nets and of (3, 4)-nets with only double and triple points is given. Up to lattice isomorphism, there are exactly 3 effective possibilities in each case, and some of these provide new examples of pencil-type line arrangements. For arrangements consisting of  14 lines and having points of multiplicity  5, we show that the non-triviality of the monodromy on the first cohomology H 1 (F) of the associated Milnor fiber F implies the arrangement is of reduced pencil-type. In particular, the monodromy is determined by the combinatorics in such cases.

Using several numerical invariants, we study a partition of the space of line arrangements in the complex projective plane, given by the intersection lattice types. We offer also a new characterization of the free plane curves using the Castelnuovo-Mumford regularity of the associated Milnor/Jacobian algebra.

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