Seven combinatorial problems around isolated quasihomogeneous singularities

Abstract

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.

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

Fig. 1
Fig. 2
Fig. 3
Fig. 4

References

  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)

    Google 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)

    MathSciNet  Article  Google 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)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Hertling, C., Kurbel, R.: Tables of weight systems of quasihomogeneous singularities.(15.08.2011) On the homepage: hilbert.math.uni-mannheim.de/CQS-homepage/index.html

  8. 8.

    Kouchnirenko, A.G.: Polyèdres de Newton et nombres de Milnor. Invent. Math. 32, 1–31 (1976)

    MathSciNet  Article  Google 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)

  10. 10.

    Kreuzer, M., Skarke, H.: On the classification of quasihomogeneous functions. Comm. Math. Phys. 150, 137–147 (1992)

    MathSciNet  Article  Google 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)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Milnor, J.: Singular Points of Complex Hypersurfaces. Ann. of Math. Stud. 61. Princeton University Press, Princeton (1968)

  13. 13.

    Milnor, J., Orlik, P.: Isolated singularities defined by weighted homogeneous polynomials. Topology 9, 385–393 (1970)

    MathSciNet  Article  Google 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)

  16. 16.

    Orlik, P., Randell, R.: The monodromy of weighted homogeneous singularities. Invent. Math. 39, 199–211 (1977)

    MathSciNet  Article  Google Scholar 

  17. 17.

    Saito, K.: Quasihomogene isolierte Singularitäten von Hyperflächen. Invent. Math. 14, 123–142 (1971)

    MathSciNet  Article  Google 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), 479–526

  19. 19.

    Saito, K.: On the existence of exponents prime to the Coxeter number. J. Algebra 114(2), 333–356 (1988)

    MathSciNet  Article  Google Scholar 

  20. 20.

    Sebastiani, M., Thom, R.: Un résultat sur la monodromie. Invent. Math. 13, 90–96 (1971)

    MathSciNet  Article  Google 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)

    Article  Google 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. 277–300

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Claus Hertling.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

This work was supported by the DFG Grant He2287/4-1 (SISYPH).

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Hertling, C., Zilke, P. Seven combinatorial problems around isolated quasihomogeneous singularities. J Algebr Comb 50, 447–482 (2019). https://doi.org/10.1007/s10801-018-0864-9

Download citation

Keywords

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

Mathematics Subject Classification

  • 32S40
  • 12Y05
  • 05C22
  • 05C25