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.
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References
Anick D.J.: On the homology of associative algebras. Trans. Amer. Math. Soc. 296(2), 641–659 (1986)
Backelin J., Fröberg R.: Koszul algebras, Veronese subrings and rings with linear resolutions. Rev. Roumaine Math. Pures Appl. 30(2), 85–97 (1985)
Berget A.: Products of linear forms and Tutte polynomials. European J. Combin. 31(7), 1924–1935 (2010)
Björner, A.: The homology and shellability of matroids and geometric lattices. In: White, N. (ed.) Matroid Applications, Encyclopedia Math. Appl., Vol. 40, pp. 226–283. Cambridge Univ. Press, Cambridge (1992)
Björner A., Ziegler G.M.: Broken circuit complexes: factorizations and generalizations. J. Combin. Theory Ser. B 51(1), 96–126 (1991)
Brylawski T.: Modular constructions for combinatorial geometries. Trans. Amer. Math. Soc. 203, 1–44 (1975)
Brylawski T.: The broken-circuit complex. Trans. Amer. Math. Soc. 234(2), 417–433 (1977)
Brylawski T., Oxley J.: The broken-circuit complex: its structure and factorizations. European J. Combin. 2(2), 107–121 (1981)
Cartan, H., Eilenberg, S.: Homological Algebra. Princeton University Press, Princeton, NJ (1999)
Denham G., Suciu A.I.: On the homotopy Lie algebra of an arrangement. Michigan Math. J. 54(2), 319–340 (2006)
Denham, G., Suciu, A.I.: Multinets, parallel connections, and Milnor fibrations of arrangements. Proc. London Math. Soc., doi:10.1112/plms/pdt058, to appear
Denham G., Yuzvinsky S.: Annihilators of Orlik-Solomon relations. Adv. Appl. Math. 28(2), 231–249 (2002)
Eisenbud, D.: Commutative Algebra. Grad. Texts in Math., Vol. 150. Springer-Verlag, New York (1995)
Falk M.: Line-closed matroids, quadratic algebras, and formal arrangements. Adv. in Appl. Math. 28(2), 250–271 (2002)
Falk, M.J., Proudfoot, N.J.: Parallel connections and bundles of arrangements. Topology Appl. 118(1-2), 65–83 (2002) (1999).
Grayson, D., Stillman, M., Eisenbud, D.: Macaulay2—a software system for algebraic geometry and commutative algebra. Available online at http://www.math.uiuc.edu/Macaulay2
Horiuchi H., Terao H.: The Poincaré series of the algebra of rational functions which are regular outside hyperplanes. J. Algebra 266(1), 169–179 (2003)
Huh J., Katz E.: Log-concavity of characteristic polynomials and the Bergman fan of matroids. Math. Ann. 354(3), 1103–1116 (2012)
Jambu M., Papadima Ş: A generalization of fiber-type arrangements and a new deformation method. Topology 37(6), 1135–1164 (1998)
Jambu M., Papadima Ş.: Deformations of hypersolvable arrangements. Topology Appl. 118(1-2), 103–111 (2002)
Jensen C., McCammond J., Meier J.: The integral cohomology of the group of loops. Geom. Topol. 10, 759–784 (2006)
Kung J.P.S.: Sign-coherent identities for characteristic polynomials of matroids. Combin. Probab. Comput. 2(1), 33–51 (1993)
Looijenga E.: Compactifications defined by arrangements. I: the ball quotient case. Duke Math. J. 118(1), 151–187 (2003)
Orlik, P., Terao, H.: Arrangements of Hyperplanes. Grundlehren Math. Wiss., Vol. 300. Springer-Verlag, Berlin (1992)
Orlik P., Terao H.: Commutative algebras for arrangements. Nagoya Math. J. 134, 65–73 (1994)
Oxley, J.: Matroid Theory. Second edition. Oxf. Grad. Texts Math., Vol. 21. Oxford University Press, Oxford (2011)
Paris L.: Intersection subgroups of complex hyperplane arrangements. Topology Appl. 105(3), 319–343 (2000)
Polishchuk, A., Positselski, L.: Quadratic Algebras. Univ. Lecture Ser., Vol. 37. American Mathematical Society, Providence, RI (2005)
Proudfoot N., Speyer D.: A broken circuit ring. Beiträge Algebra Geom. 47(1), 161–166 (2006)
Sanyal R., Sturmfels B., Vinzant C.: The entropic discriminant. Adv. Math. 244, 678–707 (2013)
Schenck H.: Resonance varieties via blowups of \({\mathbb{P}^2}\) and scrolls. Int. Math. Res. Not. 2011(20), 4756–4778 (2011)
Schenck H., Tohǎneanu Ş.O.: The Orlik-Terao algebra and 2-formality. Math. Res. Lett. 16(1), 171–182 (2009)
Shelton, B., Yuzvinsky, S.: Koszul algebras from graphs and hyperplane arrangements. J. London Math. Soc. (2) 56(3), 477–490 (1997)
Stanley R.P.: Modular elements of geometric lattices. Algebra Universalis 1, 214–217 (1971/72)
Terao H.: Modular elements of lattices and topological fibration. Adv. Math. 62(2), 135–154 (1986)
Terao H.: Algebras generated by reciprocals of linear forms. J. Algebra 250(2), 549–558 (2002)
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Denham, G., Garrousian, M. & Tohǎneanu, Ş.O. Modular Decomposition of the Orlik-Terao Algebra. Ann. Comb. 18, 289–312 (2014). https://doi.org/10.1007/s00026-014-0223-z
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DOI: https://doi.org/10.1007/s00026-014-0223-z