Modular Decomposition of the Orlik-Terao Algebra

Let ${mathcal{A}}$ be a collection of n linear hyperplanes in ${mathbb{k}^ell}$ , where ${mathbb{k}}$ is an algebraically closed field. The Orlik-Terao algebra of ${mathcal{A}}$ is the subalgebra ${{rm R}(mathcal{A})}$ of the rational functions generated by reciprocals of linear forms vanishing on hyperplanes of ${mathcal{A}}$ . It determines an irreducible subvariety ${Y (mathcal{A})}$ of ${mathbb{P}^{n-1}}$ . We show that a flat X of ${mathcal{A}}$ is modular if and only if ${{rm R}(mathcal{A})}$ is a split extension of the Orlik-Terao algebra of the subarrangement ${mathcal{A}_X}$ . This provides another refinement of Stanley’s modular factorization theorem [34] and a new characterization of modularity, similar in spirit to the fibration theorem of [27]. We deduce that if ${mathcal{A}}$ 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.