Closing two recent conjectures related to the Jacobian ideal of hyperplane arrangements

Abstract

This work is about two conjectures stated by Burity--Simis--Tohăneanu regarding the Jacobian ideal of the defining polynomial of a central arrangement of $m$ hyperplanes. One settles one of these conjectures referring to the Jacobian ideal being a minimal reduction of the ideal of $(m-1)$-fold products. The second conjecture claiming the linear type property of the Jacobian ideal is disproved in rank at least four, by means of an explicit counter-example. In the latter the corresponding Rees algebra admits a torsion defining equation which is a Pfaffian syzygetic obstruction in degree two. One also relates this Pfaffian obstruction to circuits and codimension-two flats of the arrangement.