The geometry of zonotopal algebras II: Orlik--Terao algebras and Schubert varieties

Zonotopal algebras, introduced by Postnikov–Shapiro–Shapiro, Ardila–Postnikov, and Holtz–Ron, show up in many different contexts, including approximation theory, representation theory, Donaldson–Thomas theory, and hypertoric geometry. In the first half of this paper, we construct a perfect pairing between the internal zonotopal algebra of a linear space and the reduced Orlik–Terao algebra of the Gale dual linear space. As an application, we prove a conjecture of Moseley–Proudfoot–Young that relates the reduced Orlik–Terao algebra of a graph to the cohomology of a certain configuration space. In the second half of the paper, we interpret the Macaulay inverse system of a zonotopal algebra as the space of sections of a sheaf on the Schubert variety of a linear space. As an application of this, we use an equivariant resolution of the structure sheaf of the Schubert variety inside of a product of projective lines to produce an exact sequence relating internal and external zonotopal algebras.