Seven combinatorial problems around isolated quasihomogeneous singularities

  • Claus Hertling
  • Philip Zilke


This paper proposes seven combinatorial problems around formulas for the characteristic polynomial and the spectral numbers of an isolated quasihomogeneous hypersurface singularity. One of them is a new conjecture on the characteristic polynomial. It is an amendment to an old conjecture of Orlik on the integral monodromy of an isolated quasihomogeneous singularity. The search for a combinatorial proof of the new conjecture led us to the seven purely combinatorial problems.


Isolated quasihomogeneous singularity Weight system Monodromy Characteristic polynomial Combinatorial problems Orlik blocks 

Mathematics Subject Classification

32S40 12Y05 05C22 05C25 



  1. 1.
    Aigner, M.: Combinatorial Theory. Grundlehren der Math. Wiss. 234. Springer, Berlin (1979)Google Scholar
  2. 2.
    Arnold, V.I., Gusein-Zade, S.M., Varchenko, A.N.: Singularities of Differentiable Maps, vol. I. Birkhäuser, Boston (1985)CrossRefGoogle Scholar
  3. 3.
    Hertling, C.: Brieskorn lattices and Torelli type theorems for cubics in \(\mathbb{P}^{3}\) and for Brieskorn–Pham singularities with coprime exponents. Singularities, the Brieskorn anniversary volume. Progress in Mathematics, vol. 162, pp. 167–194. Birkhäuser Verlag, Basel (1998)Google Scholar
  4. 4.
    Hertling, C.: \(\mu \)-constant monodromy groups and marked singularities. Ann. Inst. Fourier (Grenoble) 61(7), 2643–2680 (2011)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Hertling, C.: Automorphisms with eigenvales in \(S^1\) of a \(\mathbb{Z}\)-lattice with cyclic finite monodromy. preprint, arXiv:1801.07924.pdf, 33 p. (24.01.2018)
  6. 6.
    Hertling, C., Kurbel, R.: On the classification of quasihomogeneous singularities. J. Singul. 4, 131–153 (2012)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Hertling, C., Kurbel, R.: Tables of weight systems of quasihomogeneous singularities.(15.08.2011) On the homepage: Scholar
  8. 8.
    Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1–31 (1976)MathSciNetCrossRefGoogle Scholar
  9. 9.
    Kouchnirenko, A.G.: Criteria for the existence of a non-degenerate quasihomogeneous function with given weights. Uspekhi Mat. Nauk 32(3), 169–170 (1977). (In Russian)Google Scholar
  10. 10.
    Kreuzer, M., Skarke, H.: On the classification of quasihomogeneous functions. Comm. Math. Phys. 150, 137–147 (1992)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Michel, F., Weber, C.: Sur le rôle de la monodromie entière dans la topologie des singularités. Ann. Inst. Fourier (Grenoble) 36, 183–218 (1986)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Milnor, J.: Singular Points of Complex Hypersurfaces. Ann. of Math. Stud. 61. Princeton University Press, Princeton (1968)Google Scholar
  13. 13.
    Milnor, J., Orlik, P.: Isolated singularities defined by weighted homogeneous polynomials. Topology 9, 385–393 (1970)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Orlik, P.: On the homology of weighted homogeneous manifolds. In: Lecture Notes in Math. 298, Springer, Berlin, pp. 260–269 (1972)Google Scholar
  15. 15.
    Orlik, P., Randell, R.: The classification and monodromy of weighted homogeneous singularities. Preprint, 40 p. (1976 or 1977)Google Scholar
  16. 16.
    Orlik, P., Randell, R.: The monodromy of weighted homogeneous singularities. Invent. Math. 39, 199–211 (1977)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Saito, K.: Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14, 123–142 (1971)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Saito, K.: Regular systems of weights and their associated singularities. In: Complex Analytic Singularities. Advanced Studies in Pure Math. 8, Kinokuniya & North Holland (1987), 479–526Google Scholar
  19. 19.
    Saito, K.: On the existence of exponents prime to the Coxeter number. J. Algebra 114(2), 333–356 (1988)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Sebastiani, M., Thom, R.: Un résultat sur la monodromie. Invent. Math. 13, 90–96 (1971)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Shcherbak, O.P.: Conditions for the existence of a non-degenerate mapping with a given support. Funct. Anal. Appl. 13, 154–155 (1979)CrossRefGoogle Scholar
  22. 22.
    Wall, C.T.C.: Weighted homogeneous complete intersections. In: Algebraic geometry and singularities (La Rábida, 1991). Progr. Math. 134, Birkhäuser, Basel (1996), pp. 277–300Google Scholar

Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Lehrstuhl für Mathematik VIUniversität MannheimMannheimGermany

Personalised recommendations