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

We describe several relations between homological invariant of characters of fundamental groups of smooth projective varieties called depth and maps onto orbicurves. This extends previously studied relations between families of local systems and holomorphic maps onto hyperbolic curves. Firstly we derive existence of characters of the depth bounded below by the number of independent orbifold pencils and conversly deduces for some class of characters existence of the several independent pencils which number is equal to the depth of the character. Secondly, we give a new relation between depth of characters of the fundamental group and solutions of Diophantine equation over field of rational functions related to the Pell equation. Finally we give a Hodge theoretical characterization of essential coordinate characters of the fundamental groups of the complements to plane curves i.e. characters existence of which cannot be detected by considering homology of branched abelian covers. cover.

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