Mathematical Research Letters

Volume 16 (2009)

Number 1

The Orlik-Terao algebra and 2-formality

Pages: 171 – 182

DOI: http://dx.doi.org/10.4310/MRL.2009.v16.n1.a17

Authors

\c Stefan O. Toh\v aneanu (University of Cincinnati)

Hal Schenck (University of Illinois)

Abstract

The Orlik-Solomon algebra is the cohomology ring of the complement of a hyperplane arrangement $\A \subseteq \C^n$; it is the quotient of an exterior algebra $\Lambda(V)$ on $|\A|$ generators. In \cite{ot1}, Orlik and Terao introduced a commutative analog $Sym(V^*)/I$ of the Orlik-Solomon algebra to answer a question of Aomoto and showed the Hilbert series depends only on the intersection lattice $L(\A)$. In \cite{fr}, 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 $I_2$ of the Orlik-Terao ideal $I$. The key is that 2-formality is determined by the tangent space $T_p(V(I_2))$ at a generic point $p$.

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