Depth of cohomology support loci for quasi-projective varieties via orbifold pencils

We describe several relations between a homological invariant of characters of fundamental groups of projective manifolds called depth and maps onto orbicurves. This extends previously studied relations between families of local systems and holomorphic maps onto hyperbolic curves. First, we derive the existence of characters whose depth is bounded below by the number of independent orbifold pencils. Conversely, for some class of characters, we deduce the existence of as many independent pencils as the depth of the character. Second, we show a new relation between depth of characters of the fundamental group and solutions of a certain Diophantine equation (related to the Pell equation) over the field of rational functions. Finally we give a Hodge theoretical characterization of essential coordinate characters of the fundamental groups of the complements to plane curves, i.e., characters whose existence cannot be detected by considering the homology of branched abelian covers.