Orbifold Groups, Quasi-Projectivity and Covers

Abstract

Abstract. We discuss properties of complex algebraic orbifold groups, their characteristic varieties, and their abelian covers. In particular, we deal with the question of (quasi)-projectivity of orbifold groups. We also prove a structure theorem for the variety of characters of normalcrossing quasi-projective orbifold groups. Finally, we extend Sakuma's formula for the first Betti number of abelian covers of orbifold fundamental groups. Several examples are presented, including a compact orbifold group which is not projective and a Zariski pair of plane curves in P 2 that can be told by considering an unbranched cover of P 2 with an orbifold structure.

Torsion divisors of plane curves with maximal flexes and Zariski pairs

“…For the other curves, we are going to consider the orbifold groups, see [2] for definitions and properties. We may use a naïve definition as follows.…”

“…Let 𝑂 = [1 ∶ −1 ∶ 0] and 𝐿 𝑂 ∶ 𝑥 + 𝑦 = 0 and consider the group action on 𝐶 with 𝑂 being the neutral element. Let 2 , 0] and 𝐿 ⟨2⟩𝑇 1 ∶ 𝑥 + 𝜔𝑦 = 0. The intersection point 𝐿 𝑇 1 ∩ 𝐿 ⟨2⟩𝑇 1 = [0 ∶ 0 ∶ 1] cannot lie on any triangle  as the set of lines passing through [0 ∶ 0 ∶ 1] and tangent to 𝐶 is {𝐿 𝑂 , 𝐿 𝑇 1 , 𝐿 ⟨2⟩𝑇 1 }.…”