The Orlik-Terao algebra and 2-formality

TL;DR

This paper explores the relationship between 2-formality of hyperplane arrangements and the Orlik-Terao algebra, providing a criterion based on the quadratic component of the Orlik-Terao ideal.

Contribution

It establishes a necessary and sufficient condition for 2-formality using the tangent space of the quadratic component of the Orlik-Terao ideal.

Findings

2-formality is characterized by the tangent space at a generic point.

The quadratic component I_2 of the Orlik-Terao ideal determines 2-formality.

The paper links topological properties to algebraic structures in hyperplane arrangements.

Abstract

The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement A in C^n; it is the quotient of an exterior algebra E(V) on |A| generators. Orlik and Terao introduced a commutative analog S(V)/I of the Orlik-Solomon algebra to answer a question of Aomoto and showed the Hilbert series depends only on the intersection lattice L(A). Motivated by topological considerations, Falk and Randell introduced the property of 2-formality; 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 I_2 of the Orlik-Terao ideal I: 2-formality is determined by the tangent space T_p(V(I_2)) at a generic point p.