Multinets, parallel connections, and Milnor fibrations of arrangements

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.

38 pages, 8 figures; accepted for publication in the Proceedings of the London Mathematical Society

References in corpus (1)

• On combinatorial invariance of the cohomology of Milnor fiber of arrangements and Catalan equation over function field

Cited by in corpus (15)

• The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy
• Combinatorial covers and vanishing of cohomology
• Around the tangent cone theorem
• Double coverings of arrangement complements and $2$-torsion in Milnor fiber homology
• Abelian duality and propagation of resonance
• Modular decomposition of the Orlik-Terao algebra of a hyperplane arrangement
• Naturality properties and comparison results for topological and infinitesimal embedded jump loci
• The Only Complex 4-Net Is the Hesse Configuration
• Torsion in the homology of Milnor fibers of hyperplane arrangements
• Milnor fibrations of arrangements with trivial algebraic monodromy
• Sigma-invariants and tropical varieties
• Pencils of Conic-Line Curves
• The homology groups of finite cyclic covering of line arrangement complement
• The $\text{nbc}$ minimal complex of supersolvable arrangements
• $L^2$-type invariants and cohomology jump loci for complex smooth quasi-projective varieties