Projective duality of arrangements with quadratic logarithmic vector fields

TL;DR

This paper classifies rank 3 hyperplane arrangements with quadratic logarithmic derivations, revealing their structure through projective duality and expanding understanding beyond linear derivations.

Contribution

It provides a complete classification of arrangements with quadratic logarithmic derivations, extending the theory from linear to quadratic cases.

Findings

Classified rank 3 arrangements with quadratic derivations.

Linked arrangement structure to the variety of dual points.

Established a computational approach for such classifications.

Abstract

In these notes we study hyperplane arrangements having at least one logarithmic derivation of degree two that is not a combination of degree one logarithmic derivations. It is well-known that if a hyperplane arrangement has a linear logarithmic derivation not a constant multiple of the Euler derivation, then the arrangement decomposes as the direct product of smaller arrangements. The next natural step would be to study arrangements with non-trivial quadratic logarithmic derivations. On this regard, we present a computational lemma that leads to a full classification of hyperplane arrangements of rank 3 having such a quadratic logarithmic derivation. These results come as a consequence of looking at the variety of the points dual to the hyperplanes in such special arrangements.