Multinets, resonance varieties, and pencils of plane curves

TL;DR

This paper characterizes when line arrangements in the complex projective plane support nontrivial resonance varieties, linking multinets to pencils of plane curves and providing numerical conditions for their structure.

Contribution

It establishes a precise equivalence between multinets and pencils of plane curves with specific properties, advancing the understanding of resonance varieties in algebraic geometry.

Findings

Resonance varieties correspond to multinets in line arrangements.

Multinets are equivalent to certain pencils of plane curves with multiple line products.

Numerical conditions restrict the structure and multiplicities in multinets.

Abstract

We show that a line arrangement in the complex projective plane supports a nontrivial resonance variety if and only if it is the underlying arrangement of a "multinet," a multi-arrangement with a partition into three or more equinumerous classes which have equal multiplicities at each inter-class intersection point, and satisfy a connectivity condition. We also prove that this combinatorial structure is equivalent to the existence of a pencil of plane curves, also satisfying a connectivity condition, whose singular fibers include at least three products of lines, which comprise the arrangement. We derive numerical conditions which impose restrictions on the number of classes, and the line and point multiplicities that can appear in multinets, and allow us to detect whether the associated pencils yield nonlinear fiberings of the complement.