Orbifold groups, quasi-projectivity and covers

TL;DR

This paper investigates complex algebraic orbifold groups, their characteristic varieties, and abelian covers, addressing their (quasi)-projectivity, and providing new structure theorems and formulas with illustrative examples.

Contribution

It introduces a structure theorem for character varieties of orbifold groups and extends Sakuma's formula for Betti numbers of abelian covers.

Findings

Identified conditions under which orbifold groups are (quasi)-projective.

Extended Sakuma's formula to orbifold fundamental groups.

Provided examples distinguishing orbifold groups via covers.

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 normal-crossing 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 projective curves that can be told by considering an unbranched cover of the projective plane with an orbifold structure.