Multinets, parallel connections, and Milnor fibrations of arrangements

TL;DR

This paper explores the relationship between characteristic varieties, multinets, and Milnor fibrations in arrangements, providing a combinatorial method to detect torsion in the homology of Milnor fibers.

Contribution

It introduces a new combinatorial framework using multinets and polarization to identify arrangements with torsion in Milnor fiber homology.

Findings

Characteristic varieties vary with field characteristic.

A polarization construction detects torsion in Milnor fiber homology.

A combinatorial method produces arrangements with torsion in homology.

Abstract

The characteristic varieties of a space are the jump loci for homology of rank 1 local systems. The way in which the geometry of these varieties may vary with the characteristic of the ground field is reflected in the homology of finite cyclic covers. We exploit this phenomenon to detect torsion in the homology of Milnor fibers of projective hypersurfaces. One tool we use is the interpretation of the degree 1 characteristic varieties of a hyperplane arrangement complement in terms of orbifold fibrations and multinets on the corresponding matroid. Another tool is a polarization construction, based on the parallel connection operad for matroids. Our main result gives a combinatorial machine for producing arrangements whose Milnor fibers have torsion in homology.