Logarithmic derivations associated to line arrangements

TL;DR

This paper classifies rank 3 line arrangements in the projective plane over characteristic zero fields that admit a minimal logarithmic derivation of degree 3, providing explicit defining polynomials and criteria.

Contribution

It offers a complete classification of such arrangements and analyzes the structure of their minimal logarithmic derivations, which was previously unexplored.

Findings

Explicit classification of arrangements with degree 3 derivations

Identification of defining polynomials for these arrangements

Criteria for the existence of cubic minimal logarithmic derivations

Abstract

In this paper we give full classification of rank 3 line arrangements in $\mathbb P^2$ (over a field of characteristic 0) that have a minimal logarithmic derivation of degree 3. The classification presents their defining polynomials, up to a change of variables, with their corresponding affine pictures. We also analyze the shape of such a logarithmic derivation, towards obtaining criteria for a line arrangement to possess a cubic minimal logarithmic derivation.