  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.
