Annals of Combinatorics

, Volume 18, Issue 2, pp 289–312

Modular Decomposition of the Orlik-Terao Algebra

  • Graham Denham
  • Mehdi Garrousian
  • Ştefan O. Tohǎneanu
Article

DOI: 10.1007/s00026-014-0223-z

Cite this article as:
Denham, G., Garrousian, M. & Tohǎneanu, Ş.O. Ann. Comb. (2014) 18: 289. doi:10.1007/s00026-014-0223-z

Abstract

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.

Mathematics Subject Classification

52C35 16S37 13C40 05B35 13D40 

Keywords

hyperplane arrangement Orlik-Terao algebra broken circuit complex complete intersection Koszul algebra 

Copyright information

© Springer Basel 2014

Authors and Affiliations

  • Graham Denham
    • 1
  • Mehdi Garrousian
    • 2
  • Ştefan O. Tohǎneanu
    • 3
  1. 1.Department of MathematicsThe University of Western OntarioLondonCanada
  2. 2.Department of Mathematics and StatisticsUniversity of WindsorWindsorCanada
  3. 3.Department of MathematicsUniversity of IdahoMoscowUSA

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