Characters of fundamental groups of curve complements and orbifold pencils

Abstract

The present work is a user's guide to the results of [6], 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.

Complements to ample divisors and Singularities

“…3.13) of singularities in terms of polytopes and ideals of quasi-adjunction are given in [39]. For results on zero dimensional components of characteristic varieties we refer to [19], [20]. We will finish this section with an example of calculation on a large class of surfaces generalizing 6-cuspidal sextic of Zariski.…”

“…This shows that 1 6 ∈ [0, 1] is the contributing face of quasi-adjunction and now the claim about the Alexander polynomial follows from the Theorem 4.18. Note that this example also can be analyzed using methods of orbifold pencils discussed in [19], [20], [21]. .…”