Pencils of plane curves and characteristic varieties

TL;DR

This paper explores the geometric relationship between characteristic varieties of plane curve complements and pencils of curves, extending previous work on line arrangements to more general plane curves, providing new insights into their structure.

Contribution

It introduces a geometric approach linking characteristic varieties to pencils of plane curves, generalizing prior results from line arrangements to broader curve configurations.

Findings

New geometric interpretation of characteristic varieties

Relation between translated components and curve multiplicities

Extension of previous results to general plane curves

Abstract

We give a geometric approach to the relation between the irreducible components of the characteristic varieties of local systems on a plane curve arrangement complement and the associated pencils of plane curves discovered recently by M. Falk and S. Yuzvinsky in the case of line arrangements, see mathAG/0603166. In this case, this geometric point of view was already hinted to by A. Libgober and S. Yuzvinsky. Our study yields new geometric insight on the translated components of the characteristic varieties, relating them to the multiplicities of curves in the associated pencil, in close analogy to the compact situation treated by A. Beauville.