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Configuration Spaces pp 41–66Cite as

Higher Resonance Varieties of Matroids

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Part of the book series: Springer INdAM Series ((SINDAMS,volume 14))

Abstract

We present some new results about the resonance varieties of matroids and hyperplane arrangements. Though these have been the objects of ongoing study, most work so far has focused on cohomological degree 1. We show that certain phenomena become apparent only by considering all degrees at once.

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References

  1. A. Aramova, L.L. Avramov, J. Herzog, Resolutions of monomial ideals and cohomology over exterior algebras. Trans. Am. Math. Soc. 352 (2), 579–594 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  2. D. Arapura, Geometry of cohomology support loci for local systems. I. J. Algebr. Geom. 6 (3), 563–597 (1997)

    MathSciNet  MATH  Google Scholar 

  3. C. Bibby, M. Falk, I. Williams, Decomposable cocycles for p-generic arrangements, in preparation

    Google Scholar 

  4. N. Budur, Complements and higher resonance varieties of hyperplane arrangements. Math. Res. Lett. 18 (5), 859–873 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  5. H. Cartan, S. Eilenberg, Homological Algebra. Princeton Landmarks in Mathematics (Princeton University Press, Princeton, 1999); With an appendix by David A. Buchsbaum, Reprint of the 1956 original

    Google Scholar 

  6. D.C. Cohen, Triples of arrangements and local systems. Proc. Am. Math. Soc. 130 (10), 3025–3031 (2002) (electronic)

    Google Scholar 

  7. D.C. Cohen, A. Dimca, P. Orlik, Nonresonance conditions for arrangements. Ann. Inst. Fourier (Grenoble) 53 (6), 1883–1896 (2003)

    Google Scholar 

  8. D.C. Cohen, G. Denham, M. Falk, A. Varchenko, Vanishing products of one-forms and critical points of master functions. Arrangements of hyperplanes—Sapporo 2009. Advanced Studies in Pure Mathematics, vol. 62 (Mathematical Society of Japan, Tokyo, 2012), pp. 75–107

    Google Scholar 

  9. D. Cohen, G. Denham, M. Falk, H. Schenck, A. Suciu, S. Yuzvinsky, Complex Arrangements: Algebra, Geometry, Topology, in preparation

    Google Scholar 

  10. G. Denham, The combinatorial Laplacian of the Tutte complex. J. Algebr. 242 (1), 160–175 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  11. G. Denham, A. Suciu, S. Yuzvinsky, Abelian duality and propagation of resonance. arXiv:1512.07702

    Google Scholar 

  12. G. Denham, A.I. Suciu, Multinets, parallel connections, and Milnor fibrations of arrangements. Proc. Lond. Math. Soc. (3) 108 (6), 1435–1470 (2014)

    Google Scholar 

  13. A. Dimca, Singularities and Topology of Hypersurfaces. Universitext (Springer, New York, 1992)

    Book  MATH  Google Scholar 

  14. A. Dimca, Ş. Papadima, A.I. Suciu, Topology and geometry of cohomology jump loci. Duke Math. J. 148 (3), 405–457 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. D. Eisenbud, S. Popescu, S. Yuzvinsky, Hyperplane arrangement cohomology and monomials in the exterior algebra. Trans. Am. Math. Soc. 355 (11), 4365–4383 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  16. C.J. Eschenbrenner, M.J. Falk, Orlik-Solomon algebras and Tutte polynomials. J. Algebr. Combin. 10 (2), 189–199 (1999)

    Article  MathSciNet  MATH  Google Scholar 

  17. M. Falk, Arrangements and cohomology. Ann. Comb. 1 (2), 135–157 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  18. M.J. Falk, Resonance varieties over fields of positive characteristic. Int. Math. Res. Not. 2007 (3), Art. ID rnm009, 25 (2007)

    Google Scholar 

  19. M. Falk, Geometry and combinatorics of resonant weights. Arrangements, Local Systems and Singularities. Progress in Mathematics, vol. 283 (Birkhäuser, Basel, 2010), pp. 155–176

    Google Scholar 

  20. M. Falk, S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves. Compos. Math. 143 (4), 1069–1088 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  21. D. Grayson, M. Stillman, Macaulay2—a software system for algebraic geometry and commutative algebra. Available at http://www.math.uiuc.edu/Macaulay2/Citing/

  22. A. Libgober, S. Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems. Compos. Math. 121 (3), 337–361 (2000)

    Article  MathSciNet  MATH  Google Scholar 

  23. P. Orlik, L. Solomon, Combinatorics and topology of complements of hyperplanes. Invent. Math. 56 (2), 167–189 (1980)

    Article  MathSciNet  MATH  Google Scholar 

  24. P. Orlik, H. Terao, Arrangements of Hyperplanes. Grundlehren der Mathematischen Wissenschaften (Fundamental Principles of Mathematical Sciences), vol. 300 (Springer, Berlin, 1992)

    Google Scholar 

  25. J. Oxley, Matroid Theory. Oxford Graduate Texts in Mathematics, vol. 21, 2nd edn. (Oxford University Press, Oxford, 2011)

    Google Scholar 

  26. S. Papadima, A.I. Suciu, Toric complexes and Artin kernels. Adv. Math. 220 (2), 441–477 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  27. S. Papadima, A.I. Suciu, Bieri-Neumann-Strebel-Renz invariants and homology jumping loci. Proc. Lond. Math. Soc. (3) 100 (3), 795–834 (2010)

    Google Scholar 

  28. S. Papadima, A.I. Suciu, The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy (2014, preprint). arXiv:1401.0868

    Google Scholar 

  29. V.V. Schechtman, A.N. Varchenko, Arrangements of hyperplanes and Lie algebra homology. Invent. Math. 106 (1), 139–194 (1991)

    Article  MathSciNet  MATH  Google Scholar 

  30. V. Schechtman, H. Terao, A. Varchenko, Local systems over complements of hyperplanes and the Kac-Kazhdan conditions for singular vectors. J. Pure Appl. Algebra 100 (1–3), 93–102 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  31. A.I. Suciu, Around the tangent cone theorem, in Configuration Spaces: Geometry, Topology and Representation Theory. INdAM Series, vol. 14 (Springer, Berlin, 2016, to appear)

    Google Scholar 

  32. A.I. Suciu, Fundamental groups, Alexander invariants, and cohomology jumping loci. Topology of Algebraic Varieties and Singularities. Contemporary Mathematics, vol. 538 (American Mathematical Society, Providence, 2011), pp. 179–223

    Google Scholar 

  33. H. Terao, Modular elements of lattices and topological fibration. Adv. Math. 62 (2), 135–154 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  34. N. White (ed.), Theory of Matroids. Encyclopedia of Mathematics and its Applications, vol. 26 (Cambridge University Press, Cambridge, 1986)

    Google Scholar 

  35. S. Yuzvinsky, Cohomology of the Brieskorn-Orlik-Solomon algebras. Commun. Algebra 23 (14), 5339–5354 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  36. S. Yuzvinsky, Resonance varieties of arrangement complements. Arrangements of Hyperplanes—Sapporo 2009. Advanced Studies in Pure Mathematics, vol. 62 (Mathematical Society of Japan, Tokyo, 2012), pp. 553–570

    Google Scholar 

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Acknowledgements

The author would like to thank Hal Schenck for the ongoing conversations from which the main ideas for this paper emerged. This work was partially supported by a grant from NSERC of Canada.

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Authors and Affiliations

  1. Department of Mathematics, University of Western Ontario, London, ON, Canada, N6A 5B7

    Graham Denham

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  1. Graham Denham
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Correspondence to Graham Denham .

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Editors and Affiliations

  1. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Filippo Callegaro

  2. Department of Mathematics, University of Rochester, Rochester, USA

    Frederick Cohen

  3. Dipartimento di Matematica, Università di Roma La Sapienza, Roma, Italy

    Corrado De Concini

  4. Department of Mathematics, University of Bremen, Bremen, Germany

    Eva Maria Feichtner

  5. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Giovanni Gaiffi

  6. Dipartimento di Matematica, Università di Pisa, Pisa, Italy

    Mario Salvetti

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Denham, G. (2016). Higher Resonance Varieties of Matroids. In: Callegaro, F., Cohen, F., De Concini, C., Feichtner, E., Gaiffi, G., Salvetti, M. (eds) Configuration Spaces. Springer INdAM Series, vol 14. Springer, Cham. https://doi.org/10.1007/978-3-319-31580-5_2

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