Koszul Orlik--Solomon Algebras from Non-supersolvable Arrangements

Abstract: The cohomology ring of the complement of a complex hyperplane arrangement is given by its Orlik--Solomon algebra. It is known that the defining ideal of the Orlik--Solomon algebra has a quadratic Gröbner basis in the standard presentation if and only if the intersection lattice is supersolvable; such algebras are automatically Koszul. In 1997, Shelton and Yuzvinsky posed the question as to whether all Koszul Orlik--Solomon algebras arise from supersolvable arrangements. We answer this question negatively using three related constructions that produce non-supersolvable arrangements whose Orlik--Solomon algebras are Koszul. Moreover, these arrangements may be chosen to be irreducible, realizable over $\mathbb{Q}$, and of any rank $\geq 3$. Our constructions rely on a result of Falk and Proudfoot which we strengthen and generalize. In two of the three cases, we show non-supersolvability using a corrected form of a result of Ziegler regarding supersolvability of parallel connections. We also construct Koszul Orlik--Terao algebras coming from non-supersolvable arrangements.