Higher Resonance Varieties of Matroids

Chapter
Part of the Springer INdAM Series book 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.

Keywords

Hyperplane arrangement Matroid Resonance variety 

Notes

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.

References

  1. 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)MathSciNetCrossRefMATHGoogle Scholar
  2. 2.
    D. Arapura, Geometry of cohomology support loci for local systems. I. J. Algebr. Geom. 6 (3), 563–597 (1997)MathSciNetMATHGoogle Scholar
  3. 3.
    C. Bibby, M. Falk, I. Williams, Decomposable cocycles for p-generic arrangements, in preparationGoogle Scholar
  4. 4.
    N. Budur, Complements and higher resonance varieties of hyperplane arrangements. Math. Res. Lett. 18 (5), 859–873 (2011)MathSciNetCrossRefMATHGoogle Scholar
  5. 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 originalGoogle Scholar
  6. 6.
    D.C. Cohen, Triples of arrangements and local systems. Proc. Am. Math. Soc. 130 (10), 3025–3031 (2002) (electronic)Google Scholar
  7. 7.
    D.C. Cohen, A. Dimca, P. Orlik, Nonresonance conditions for arrangements. Ann. Inst. Fourier (Grenoble) 53 (6), 1883–1896 (2003)Google Scholar
  8. 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. 75–107Google Scholar
  9. 9.
    D. Cohen, G. Denham, M. Falk, H. Schenck, A. Suciu, S. Yuzvinsky, Complex Arrangements: Algebra, Geometry, Topology, in preparationGoogle Scholar
  10. 10.
    G. Denham, The combinatorial Laplacian of the Tutte complex. J. Algebr. 242 (1), 160–175 (2001)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    G. Denham, A. Suciu, S. Yuzvinsky, Abelian duality and propagation of resonance. arXiv:1512.07702Google Scholar
  12. 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. 13.
    A. Dimca, Singularities and Topology of Hypersurfaces. Universitext (Springer, New York, 1992)CrossRefMATHGoogle Scholar
  14. 14.
    A. Dimca, Ş. Papadima, A.I. Suciu, Topology and geometry of cohomology jump loci. Duke Math. J. 148 (3), 405–457 (2009)MathSciNetCrossRefMATHGoogle Scholar
  15. 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)MathSciNetCrossRefMATHGoogle Scholar
  16. 16.
    C.J. Eschenbrenner, M.J. Falk, Orlik-Solomon algebras and Tutte polynomials. J. Algebr. Combin. 10 (2), 189–199 (1999)MathSciNetCrossRefMATHGoogle Scholar
  17. 17.
    M. Falk, Arrangements and cohomology. Ann. Comb. 1 (2), 135–157 (1997)MathSciNetCrossRefMATHGoogle Scholar
  18. 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. 19.
    M. Falk, Geometry and combinatorics of resonant weights. Arrangements, Local Systems and Singularities. Progress in Mathematics, vol. 283 (Birkhäuser, Basel, 2010), pp. 155–176Google Scholar
  20. 20.
    M. Falk, S. Yuzvinsky, Multinets, resonance varieties, and pencils of plane curves. Compos. Math. 143 (4), 1069–1088 (2007)MathSciNetCrossRefMATHGoogle Scholar
  21. 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. 22.
    A. Libgober, S. Yuzvinsky, Cohomology of the Orlik-Solomon algebras and local systems. Compos. Math. 121 (3), 337–361 (2000)MathSciNetCrossRefMATHGoogle Scholar
  23. 23.
    P. Orlik, L. Solomon, Combinatorics and topology of complements of hyperplanes. Invent. Math. 56 (2), 167–189 (1980)MathSciNetCrossRefMATHGoogle Scholar
  24. 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. 25.
    J. Oxley, Matroid Theory. Oxford Graduate Texts in Mathematics, vol. 21, 2nd edn. (Oxford University Press, Oxford, 2011)Google Scholar
  26. 26.
    S. Papadima, A.I. Suciu, Toric complexes and Artin kernels. Adv. Math. 220 (2), 441–477 (2009)MathSciNetCrossRefMATHGoogle Scholar
  27. 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. 28.
    S. Papadima, A.I. Suciu, The Milnor fibration of a hyperplane arrangement: from modular resonance to algebraic monodromy (2014, preprint). arXiv:1401.0868Google Scholar
  29. 29.
    V.V. Schechtman, A.N. Varchenko, Arrangements of hyperplanes and Lie algebra homology. Invent. Math. 106 (1), 139–194 (1991)MathSciNetCrossRefMATHGoogle Scholar
  30. 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)MathSciNetCrossRefMATHGoogle Scholar
  31. 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. 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. 179–223Google Scholar
  33. 33.
    H. Terao, Modular elements of lattices and topological fibration. Adv. Math. 62 (2), 135–154 (1986)MathSciNetCrossRefMATHGoogle Scholar
  34. 34.
    N. White (ed.), Theory of Matroids. Encyclopedia of Mathematics and its Applications, vol. 26 (Cambridge University Press, Cambridge, 1986)Google Scholar
  35. 35.
    S. Yuzvinsky, Cohomology of the Brieskorn-Orlik-Solomon algebras. Commun. Algebra 23 (14), 5339–5354 (1995)MathSciNetCrossRefMATHGoogle Scholar
  36. 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. 553–570Google Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  1. 1.Department of MathematicsUniversity of Western OntarioLondonCanada

Personalised recommendations