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G\"unter Ziegler has shown in 1989 that some homological invariantsassociated with the free resolutions of Jacobian ideals of line arrangementsare not determined by combinatorics. His classical example involves hexagonsinscribed in conics. Independently, Sergey Yuzvinsky has arrived in 1993 at thesame type of line arrangements in order to show that formality is notdetermined by the combinatorics. In this note we look into the geometry of suchline arrangements, and find out an unexpected relation to the classicalPascal's Theorem. Our results give information on the minimal degree of aJacobian syzygy and on the formality of such hexagonal line arrangements ingeneral, without an explicit choice for the six vertices of the hexagon.

From Pascal's Theorem to the geometry of Ziegler's line arrangements