Skip to main content

Mordell-Weil lattices and toric decompositions of plane curves

Abstract

We extend results of Cogolludo-Agustin and Libgober relating the Alexander polynomial of a plane curve C with the Mordell–Weil rank of certain isotrivial families of jacobians over \(\mathbf {P}^2\) of discriminant C. In the second part we introduce a height pairing on the (2, 3, 6) quasi-toric decompositions of a plane curve. We use this pairing and the results in the first part of the paper to construct a pair of degree 12 curves with 30 cusps and Alexander polynomial \(t^2-t+1\), but with distinct height pairing. We use the height pairing to show that these curves from a Zariski pair.

This is a preview of subscription content, access via your institution.

References

  1. 1.

    Artal Bartolo, E., Dimca, A.: On fundamental groups of plane curve complements (2015). arXiv:1507.08178v1 (preprint)

  2. 2.

    Cogolludo-Agustín, J.-I., Libgober, A.: Mordell-Weil groups of elliptic threefolds and the Alexander module of plane curves. J. Reine Angew. Math. 697, 15–55 (2014)

    MathSciNet  MATH  Google Scholar 

  3. 3.

    Degtyarev, A.: Oka’s conjecture on irreducible plane sextics. J. Lond. Math. Soc. 2(78), 329–351 (2008)

    MathSciNet  Article  MATH  Google Scholar 

  4. 4.

    Degtyarev, A.: Oka’s conjecture on irreducible plane sextics. II. J. Knot Theory Ramif. 18, 1065–1080 (2009)

    MathSciNet  Article  MATH  Google Scholar 

  5. 5.

    Dimca, A.: Differential forms and hypersurface singularities. In: Singularity theory and its applications, Part I (Coventry, 1988/1989), vol. 1462 of Lecture Notes in Math., pp. 122-153. Springer, Berlin (1991)

  6. 6.

    Dimca, A.: Singularities and topology of hypersurfaces. Universitext. Springer, New York (1992)

    Book  MATH  Google Scholar 

  7. 7.

    Eyral, C., Oka, M.: On the fundamental groups of the complements of plane singular sextics. J. Math. Soc. Jpn. 57, 37–54 (2005)

    MathSciNet  Article  MATH  Google Scholar 

  8. 8.

    Kulikov, V.S.: Mixed Hodge structures and singularities. Cambridge tracts in mathematics, vol. 132. Cambridge University Press, Cambridge (1998)

  9. 9.

    Libgober, A.: Alexander invariants of plane algebraic curves. In: Singularities, part 2 (Arcata, Calif., 1981), vol. 40 of Proc. Sympos. Pure Math., pp. 135–143. American Mathematical Society, Providence (1983)

  10. 10.

    Libgober, A.: Position of singularities of hypersurfaces and the topology of their complements. J. Math. Sci. 82, 3194–3210 (1996) (algebraic geometry, 5)

  11. 11.

    Libgober, A.: On Mordell-Weil groups of isotrivial abelian varieties over function fields. Math. Ann. 357, 605–629 (2013)

    MathSciNet  Article  MATH  Google Scholar 

  12. 12.

    Libgober, A.: Albanese varieties of abelian covers. J. Singul. 12, 105–123 (2015)

    MathSciNet  MATH  Google Scholar 

  13. 13.

    Lindner, N.: Cuspidal plane curves of degree 12 and their Alexander polynomials. Master’s thesis, Humboldt Universität zu Berlin, Berlin (2012)

  14. 14.

    Milne, J.S.: Abelian varieties (v2.00). http://www.jmilne.org/math/ (2008)

  15. 15.

    Miranda, R.: The basic theory of elliptic surfaces. Dottorato di Ricerca in Matematica. ETS Editrice, Pisa (1989)

    Google Scholar 

  16. 16.

    Peters, C.A.M., Steenbrink, J.H.M.: Mixed Hodge structures, vol. 52 of Ergebnisse der Mathematik und ihrer Grenzgebiete, Folge, vol. 3. Springer, Berlin (2008)

  17. 17.

    Scherk, J., Steenbrink, J.H.M.: On the mixed Hodge structure on the cohomology of the Milnor fibre. Math. Ann. 271, 641–665 (1985)

    MathSciNet  Article  MATH  Google Scholar 

  18. 18.

    Shioda, T.: On the Mordell-Weil lattices. Comment. Math. Univ. St. Paul. 39, 211–240 (1990)

    MathSciNet  MATH  Google Scholar 

  19. 19.

    Varchenko, A.N.: Semicontinuity of the spectrum and an upper bound for the number of singular points of the projective hypersurface. Dokl. Akad. Nauk SSSR 270, 1294–1297 (1983)

    MathSciNet  Google Scholar 

  20. 20.

    Zariski, O.: On the linear connection index of the algebraic surfaces \(z^n=f(x, y)\). Proc. Natl. Acad. Sci. USA 15, 494–501 (1929)

    Article  MATH  Google Scholar 

Download references

Author information

Affiliations

Authors

Corresponding author

Correspondence to Remke Kloosterman.

Additional information

Remke Kloosterman would like to thank Orsola Tommasi for several comments on a previous version of this paper. Remke Kloosterman would like to thank the referee for suggesting various improvements on the presentation.

Communicated by Ngaiming Mok.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Kloosterman, R. Mordell-Weil lattices and toric decompositions of plane curves. Math. Ann. 367, 755–783 (2017). https://doi.org/10.1007/s00208-016-1399-9

Download citation

Mathematics Subject Classification

  • 14H50
  • 14H20
  • 14H30
  • 14H40
  • 14J27
  • 14J30
  • 14J70