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The present work is a user's guide to the results of a previous paper by thesecond and third authors, where a description of the space of characters of aquasi-projective variety was given in terms of global quotient orbifoldpencils.  Below we consider the case of plane curve complements and hyperplanearrangements. In particular, an infinite family of curves exhibiting charactersof any torsion and depth~3 will be discussed. Also, in the context of linearrangements, it will be shown how geometric tools, such as the existence oforbifold pencils, can replace the group theoretical computations viafundamental groups when studying characters of finite order, specially ordertwo. Finally, we revisit an Alexander-equivalent Zariski pair considered in theliterature and show how the existence of such pencils distinguishes bothcurves.

Depth of characters of curve complements and orbifold pencils