Characteristic varieties of algebraic curves

TL;DR

This paper investigates the tori associated with the fundamental groups of algebraic plane curves, revealing their role in understanding homology of finite abelian covers and connecting them to singularities and linear systems.

Contribution

It introduces a method to compute these tori using linear systems derived from curve singularities, extending understanding of algebraic curve complements.

Findings

Tori encode homology information of branched covers.

Explicit calculations relate tori to singularities and linear systems.

Connections established between tori, arrangement lattices, and Aomoto complexes.

Abstract

We study tori attached to the fundamental groups of plane curves with arbitrary singularities. These tori provide complete information about homology of finite abelian covers of the plane branched along the curve. We calculate these tori in terms of certain linear systems determined by the singularities of the curve. In the case of the complements to a union of lines they can be calculated from the lattice of the arrangement and are closely related to the components of the space of Aomoto complexes with prescribed homology.