Depth of characters of curve complements and orbifold pencils

The present work is a user's guide to the results of a previous paper by the second and third authors, where a description of the space of characters of a quasi-projective variety was given in terms of global quotient orbifold pencils. Below we consider the case of plane curve complements and hyperplane arrangements. In particular, an infinite family of curves exhibiting characters of any torsion and depth~3 will be discussed. Also, in the context of line arrangements, it will be shown how geometric tools, such as the existence of orbifold pencils, can replace the group theoretical computations via fundamental groups when studying characters of finite order, specially order two. Finally, we revisit an Alexander-equivalent Zariski pair considered in the literature and show how the existence of such pencils distinguishes both curves.