Motivic Milnor Fibers of Plane Curve Singularities



We compute the motivic Milnor fiber of a complex plane curve singularity in an inductive and combinatoric way using the extended simplified resolution graph. The method introduced in this article has a consequence that one can study the Hodge–Steenbrink spectrum of such a singularity in terms of that of a quasi-homogeneous singularity.


Plane curve singularity Newton polyhedron Resolution of singularity Extended resolution graph Arc spaces Motivic integration Motivic zeta function Motivic Milnor fiber 

Mathematics Subject Classification (2010)

Primary 14B05 14E15 14E18 14H20 14M25 32B30 32S55 



This research is funded by the Vietnam National Foundation for Science and Technology Development (NAFOSTED) under grant number FWO.101.2015.02.

The author would like to thank The Abdus Salam International Centre for Theoretical Physics (ICTP), The Vietnam Institute for Advanced Study in Mathematics (VIASM), and Department of Mathematics - KU Leuven for warm hospitality during his visits.


  1. 1.
    A’Campo, N., Oka, M.: Geometry of plane curves via Tschirnhausen resolution tower. Osaka J. Math. 33, 1003–1033 (1996)MathSciNetMATHGoogle Scholar
  2. 2.
    Budur, N.: On Hodge spectrum and multiplier ideals. Math. Ann. 327, 257–270 (2003)MathSciNetCrossRefMATHGoogle Scholar
  3. 3.
    Denef, J., Loeser, F.: Motivic Igusa zeta functions. J. Algebraic Geom. 7, 505–537 (1998)MathSciNetMATHGoogle Scholar
  4. 4.
    Denef, J., Loeser, F.: Germs of arcs on singular algebraic varieties and motivic integration. Invent. Math. 135, 201–232 (1999)MathSciNetCrossRefMATHGoogle Scholar
  5. 5.
    Denef, J., Loeser, F.: Motivic exponential integrals and a motivic Thom–Sebastiani theorem. Duke Math. J. 99, 285–309 (1999)MathSciNetCrossRefMATHGoogle Scholar
  6. 6.
    Denef, J., Loeser, F.: Geometry on arc spaces of algebraic varieties. In: Casacuberta, C., et al. (eds.) European Congress of Mathematics, Vol. 1 (Barcelona, 2000). Progress in Mathematics, vol. 201, pp 327–348. Basel, Birkhaüser (2001)Google Scholar
  7. 7.
    Denef, J., Loeser, F.: Lefschetz numbers of iterates of the monodromy and truncated arcs. Topology 41, 1031–1040 (2002)MathSciNetCrossRefMATHGoogle Scholar
  8. 8.
    González Villa, M., Kennedy, G., McEwan, L.: A recursive formula for the motivic Milnor fiber of a plane curve (2016). arXiv:1610.08487
  9. 9.
    Guibert, G.: Espaces d’arcs et invariants d’Alexander. Comment. Math. Helv. 77, 783–820 (2002)MathSciNetCrossRefMATHGoogle Scholar
  10. 10.
    Guibert, G., Loeser, F., Merle, M.: Iterated vanishing cycles, convolution, and a motivic analogue of a conjecture of Steenbrink. Duke Math. J. 132, 409–457 (2006)MathSciNetCrossRefMATHGoogle Scholar
  11. 11.
    Guibert, G., Loeser, F., Merle, M.: Nearby cycles and composition with a nondegenerate polynomial. Int. Math. Res. Not. 31, 1873–1888 (2005)MathSciNetCrossRefMATHGoogle Scholar
  12. 12.
    Guibert, G., Loeser, F., Merle, M.: Composition with a two variable function. Math. Res. Lett. 16, 439–448 (2009)MathSciNetCrossRefMATHGoogle Scholar
  13. 13.
    Lê, D.T., Oka, M.: On resolution complexity of plane curves. Kodai Math. J. 18, 1–36 (1995)MathSciNetCrossRefMATHGoogle Scholar
  14. 14.
    Lê, Q.T.: On a conjecture of Kontsevich and Soibelman. Algebra Number Theory 6, 389–404 (2012)MathSciNetCrossRefMATHGoogle Scholar
  15. 15.
    Lê, Q.T.: Zeta function of degenerate plane curve singularity. Osaka J. Math. 49, 687–697 (2012)MathSciNetMATHGoogle Scholar
  16. 16.
    Lê, Q.T.: The motivic Thom–Sebastiani theorem for regular and formal functions. J. Reine Angew. Math. (2015). doi: 10.1515/crelle-2015-0022
  17. 17.
    Looijenga, E.: Motivic measures. Astérisque 276, 267–297 (2002). Séminaire Bourbaki 1999/2000, no. 874MathSciNetMATHGoogle Scholar
  18. 18.
    Milnor, J.: Singular Points of Complex Hypersurface. Ann. Math. Stud., vol. 61. Princeton University Press, Princeton (1968)Google Scholar
  19. 19.
    Saito, M.: On Steenbrink’s conjecture. Math. Ann. 289, 703–716 (1991)MathSciNetCrossRefMATHGoogle Scholar
  20. 20.
    Saito, M.: Exponents of an irreducible plane curve singularity (2000). arXiv:math/0009133
  21. 21.
    Steenbrink, J.H.M.: Motivic Milnor fibre for nondegenerate function germs on toric singularities. In: Ibadula, D., Veys, W. (eds.) Bridging Algebra, Geometry, and Topology. Springer Proceedings in Mathematics & Statistics, vol. 96, pp 255–267. Springer, Cham (2014)Google Scholar

Copyright information

© Vietnam Academy of Science and Technology (VAST) and Springer Science+Business Media Singapore 2017

Authors and Affiliations

  1. 1.Department of MathematicsVietnam National UniversityHanoiVietnam

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