Skip to main content
SpringerLink
Log in

Jumping coefficients and spectrum of a hyperplane arrangement

  • Published:
Mathematische Annalen Aims and scope Submit manuscript

Abstract

In an earlier version of this paper written by the second named author, we showed that the jumping coefficients of a hyperplane arrangement depend only on the combinatorial data of the arrangement as conjectured by Mustaţǎ. For this we proved a similar assertion on the spectrum. After this first proof was written, the first named author found a more conceptual proof using the Hirzebruch–Riemann–Roch theorem where the assertion on the jumping numbers was proved without reducing to that for the spectrum. In this paper we improve these methods and show that the jumping numbers and the spectrum are calculable in low dimensions without using a computer. In the reduced case we show that these depend only on fewer combinatorial data, and give completely explicit combinatorial formulas for the jumping coefficients and (part of) the spectrum in the case the ambient dimension is 3 or 4. We also give an analogue of Mustaţǎ’s formula for the spectrum.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

We’re sorry, something doesn't seem to be working properly.

Please try refreshing the page. If that doesn't work, please contact support so we can address the problem.

References

  1. Beilinson, A., Bernstein, J., Deligne, P.: Faisceaux pervers, Astérisque, vol. 100. Soc. Math. France, Paris (1982)

  2. Brieskorn, E.: Sur les groupes de tresses [d’après V.I. Arnold], Séminaire Bourbaki, 24ème année (1971/1972), Exp. No. 401, Lect. Notes in Math. vol. 317, pp. 21–44. Springer, Berlin (1973)

  3. Budur N.: On Hodge spectrum and multiplier ideals. Math. Ann. 327, 257–270 (2003)

    Article  MATH  MathSciNet  Google Scholar 

  4. Budur, N.: Jumping numbers of hyperplane arrangements. Comm. Algebra (to appear). arXiv:0802. 0878

  5. Budur, N.: Hodge spectrum of hyperplane arrangements. arXiv:0809.3443 (unpublished)

  6. Budur N., Saito M.: Multiplier ideals, V-filtration, and spectrum. J. Alg. Geom. 14, 269–282 (2005)

    MATH  MathSciNet  Google Scholar 

  7. Cohen D.C., Suciu A.: On Milnor fibrations of arrangements. J. Lond. Math. Soc. 51, 105–119 (1995)

    MATH  MathSciNet  Google Scholar 

  8. De Concini C., Procesi C.: Wonderful models of subspace arrangements. Selecta Math. (N.S.) 1, 459–494 (1995)

    Article  MATH  MathSciNet  Google Scholar 

  9. Deligne P.: Equations Différentiellesà Points Singuliers Réguliers, Lect. Notes in Math. vol. 163. Springer, Berlin (1970)

    Google Scholar 

  10. Deligne P.: Théorie de Hodge II. Publ. Math. IHES 40, 5–58 (1971)

    MATH  MathSciNet  Google Scholar 

  11. Deligne, P.: Le formalisme des cycles évanescents, in SGA7 XIII and XIV, Lect. Notes in Math. 340, Springer, Berlin, pp. 82–115 and 116–164 (1973)

  12. Dimca A.: Singularities and Topology of Hypersurfaces, Universitext. Springer, Berlin (1992)

    Google Scholar 

  13. Dimca A., Saito M.: A generalization of Griffiths’s theorem on rational integrals. Duke Math. J. 135, 303–326 (2006)

    Article  MATH  MathSciNet  Google Scholar 

  14. Ein L., Lazarsfeld R., Smith K.E., Varolin D.: Jumping coefficients of multiplier ideals. Duke Math. J. 123, 469–506 (2004)

    Article  MATH  MathSciNet  Google Scholar 

  15. Esnault H.: Fibre de Milnor d’un cône sur une courbe plane singulière. Inv. Math. 68, 477–496 (1982)

    Article  MATH  MathSciNet  Google Scholar 

  16. Esnault, H., Viehweg, E.: Revêtements cycliques, in Algebraic threefolds (Varenna, 1981). Lecture Notes in Math., vol. 947, pp. 241–250. Springer, Berlin (1982)

  17. Fulton W.: Intersection Theory. Springer, Berlin (1984)

    MATH  Google Scholar 

  18. Hartshorne R.: Algebraic Geometry. Springer, Berlin (1977)

    MATH  Google Scholar 

  19. Hirzebruch F.: Topological Method in Algebraic Geometry. Springer, Berlin (1966)

    Google Scholar 

  20. Jouanolou, J.-P.: Cohomologie de quelques schémas classiques et théorie cohomologique des classes de Chern. In: SGA5, Lecture Notes in Math. 589, pp. 282–350. Springer, Berlin (1977)

  21. Lazarsfeld, R.: Positivity in algebraic geometry II, Ergebnisse der Mathematik und ihrer Grenzgebiete. 3. Folge, A series of Modern Surveys in Mathematics, vol. 49. Springer-Verlag, Berlin (2004)

  22. Mustaţǎ M.: Multiplier ideals of hyperplane arrangements. Trans. Am. Math. Soc. 358, 5015–5023 (2006)

    Article  Google Scholar 

  23. Nadel A.M.: Multiplier ideal sheaves and Kähler-Einstein metrics of positive scalar curvature. Ann. Math. 132, 549–596 (1990)

    Article  MathSciNet  Google Scholar 

  24. Orlik P., Solomon L.: Combinatorics and topology of complements of hyperplanes. Inv. Math. 56, 167–189 (1980)

    Article  MATH  MathSciNet  Google Scholar 

  25. Rybnikov, G.: On the fundamental group of the complement of a complex hyperplane arrangement (math.AG/9805056)

  26. Saito, M.: Mixed Hodge modules. Publ. RIMS, Kyoto Univ. 26, pp. 221–333 (1990)

  27. Saito M.: Multiplier ideals, b-function, and spectrum of a hypersurface singularity. Compos. Math. 143, 1050–1068 (2007)

    MATH  MathSciNet  Google Scholar 

  28. Saito, M.: Bernstein-Sato polynomials of hyperplane arrangements (math.AG/0602527)

  29. Saito, M.: Jumping coefficients and spectrum of a hyperplane arrangement. (unpublished manuscript, 2007)

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

    Article  MATH  MathSciNet  Google Scholar 

  31. Steenbrink J.H.M.: The spectrum of hypersurface singularity. Astérisque 179–180, 163–184 (1989)

    MathSciNet  Google Scholar 

  32. Teitler Z.: A note on Mustaţǎ’s computation of multiplier ideals of hyperplane arrangements. Proc. Am. Math. Soc. 136, 1575–1579 (2008)

    Article  MATH  MathSciNet  Google Scholar 

  33. Walther U.: Bernstein-Sato polynomial versus cohomology of the Milnor fiber for generic hyperplane arrangements. Compos. Math. 141, 121–145 (2005)

    Article  MATH  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, The University of Notre Dame, Notre Dame, IN, 46556, USA

    Nero Budur

  2. RIMS, Kyoto University, Kyoto, 606-8502, Japan

    Morihiko Saito

Authors
  1. Nero Budur
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Morihiko Saito
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Nero Budur.

Additional information

N. Budur is supported by the NSF grant DMS-0700360. M. Saito is partially supported by Kakenhi 19540023.

Rights and permissions

Reprints and permissions

About this article

Cite this article

Budur, N., Saito, M. Jumping coefficients and spectrum of a hyperplane arrangement. Math. Ann. 347, 545–579 (2010). https://doi.org/10.1007/s00208-009-0449-y

Download citation

  • Received:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s00208-009-0449-y

Keywords

Search

Navigation