The orlik-terao algebra and 2-formality

The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement A ⊂ ℂⁿ it is the quotient of an exterior algebra A(V) on |A| generators. In [9], Orlik and Terao introduced a commutative analog Sym(V*)/I of the Orlik-Solomon algebra to answer a question of Aomoto and showed the Hubert series depends only on the intersection lattice L(A). In [6], Falk and Randell define the property of 2-formality; in this note we study the relation between 2-formality and the Orlik-Terao algebra. Our main result is a necessary and sufficient condition for 2-formality in terms of the quadratic component I₂ of the Orlik-Terao ideal I. The key is that 2-formality is determined by the tangent space T _p(V(I₂)) at a generic point p.