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In these notes we study hyperplane arrangements having at least onelogarithmic derivation of degree two that is not a combination of degree onelogarithmic derivations. It is well-known that if a hyperplane arrangement hasa linear logarithmic derivation not a constant multiple of the Eulerderivation, then the arrangement decomposes as the direct product of smallerarrangements. The next natural step would be to study arrangements withnon-trivial quadratic logarithmic derivations. On this regard, we present acomputational lemma that leads to a full classification of hyperplanearrangements of rank 3 having such a quadratic logarithmic derivation. Theseresults come as a consequence of looking at the variety of the points dual tothe hyperplanes in such special arrangements.

Projective duality of arrangements with quadratic logarithmic vector fields