Geometry of logarithmic derivations of hyperplane arrangements

TL;DR

This paper explores the geometric structure of logarithmic derivations of hyperplane arrangements by analyzing an ideal related to the arrangement and its dual, revealing connections to matroid theory and freeness properties.

Contribution

It introduces the ideal of pairs to study arrangements symmetrically via matroid duality and characterizes its variety as a subspace arrangement linked to cyclic flats.

Findings

Variety of the ideal of pairs forms a subspace arrangement.

Components correspond to cyclic flats of the arrangement.

Provides geometric insights into freeness and projective dimension results.

Abstract

We study the Hadamard product of the linear forms defining a hyperplane arrangement with those of its dual, which we view as generating an ideal in a certain polynomial ring. We use this ideal, which we call the ideal of pairs, to study logarithmic derivations and critical set varieties of arrangements in a way which is symmetric with respect to matroid duality. Our main result exhibits the variety of the ideal of pairs as a subspace arrangement whose components correspond to cyclic flats of the arrangement. As a corollary, we are able to give geometric explanations of some freeness and projective dimension results due to Ziegler and Kung--Schenck.