Modular decomposition of the Orlik-Terao algebra of a hyperplane arrangement

Let A be a collection of n linear hyperplanes in k^l, where k is an algebraically closed field. The Orlik-Terao algebra of A is the subalgebra R(A) of the rational functions generated by reciprocals of linear forms vanishing on hyperplanes of A. It determines an irreducible subvariety of projective space. We show that a flat X of A is modular if and only if R(A) is a split extension of the Orlik-Terao algebra of the subarrangement A_X. This provides another refinement of Stanley's modular factorization theorem and a new characterization of modularity, similar in spirit to the modular fibration theorem of Paris. We deduce that if A is supersolvable, then its Orlik-Terao algebra is Koszul. In certain cases, the algebra is also a complete intersection, and we characterize when this happens.