Skip to main content
SpringerLink
Log in

Plane quartics: the universal matrix of bitangents

  • Published:
Israel Journal of Mathematics Aims and scope Submit manuscript

Abstract

Aronhold’s classical result states that a plane quartic can be recovered by the configuration of any Aronhold systems of bitangents, i.e., special 7-tuples of bitangents such that the six points at which any sub-triple of bitangents touches the quartic do not lie on the same conic in the projective plane. Lehavi (cf. [L05]) proved that a smooth plane quartic can be explicitly reconstructed from its 28 bitangents; this result improved Aronhold’s method of recovering the curve. In a 2011 paper [PSV11] Plaumann, Sturmfels and Vinzant introduced an 8 × 8 symmetric matrix that parametrizes the bitangents of a nonsingular plane quartic. The starting point of their construction is Hesse’s result for which every smooth quartic curve has exactly 36 equivalence classes of linear symmetric determinantal representations. In this paper we tackle the inverse problem, i.e., the construction of the bitangent matrix starting from the 28 bitangents of the plane quartic, and we provide a Sage script intended for computing the bitangent matrix of a given curve.

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

Similar content being viewed by others

References

  1. S. H. Aronhold, Über den gegenseitigen Zusammenhang der 28 Doppeltangenten einer allgemeinen Kurve vierten Grades, Monatsberichte der Königlichen Preussische Akademie des Wissenschaften zu Berlin (1864), 499–523.

    Google Scholar 

  2. L. Caporaso and E. Sernesi, Recovering plane curves from their bitangents, Journal of Algebraic Geometry 12 (2003), 225–244.

    Article  MathSciNet  MATH  Google Scholar 

  3. L. Caporaso and E. Sernesi, Characterizing curves by their odd thetacharacteristics, Journal für die Reine und Angewandte Mathematik 562 (2003), 101–135.

    MathSciNet  MATH  Google Scholar 

  4. I. Dolgachev, Classical Algebraic Geometry: a Modern View, Cambridge University Press, Cambridge, 2012.

    Book  MATH  Google Scholar 

  5. I. Dolgachev and D. Ortland, Point Sets in Projective Spaces and Theta Functions, Astérisque 165 (1988).

    MATH  Google Scholar 

  6. J. Fay, On the Riemann–Jacobi formula, Nachrichten der Akademie der Wissenschaften in Göttingen. II. Mathematisch-Physikalische Klasse 5 (1979), 61–73.

    MathSciNet  MATH  Google Scholar 

  7. B. van Geemen and G. van der Geer, Kummer varieties and the moduli spaces of abelian varieties, American Journal of Mathematics 108 (1986), 615–641.

    Article  MathSciNet  MATH  Google Scholar 

  8. B. H. Gross and J. Harris, On some geometric constructions related to theta characteristics, in Contributions to Automorphic Forms, Geometry and Number Theory, John Hopkins University Press, Baltimore, MD, 2004, pp. 279–311.

    Google Scholar 

  9. S. Grushevsky and R. Salvati Manni, Gradients of odd theta functions, Journal für die Reine und Angewandte Mathematik 573 (2004), 45–59.

    Article  MathSciNet  MATH  Google Scholar 

  10. J. Guàrdia, On the Torelli problem and Jacobian nullwerte in genus three, Michigan Mathematical Journal 60 (2011), 51–65.

    Article  MathSciNet  MATH  Google Scholar 

  11. O. Hesse, Über die Doppeltangenten der Curven vierter Ordnung, Journal für die Reine und Angewandte Mathematik 49 (1855), 279–332.

    Article  MathSciNet  Google Scholar 

  12. J.-I. Igusa, Theta Functions, Die Grundlehren der Mathematischen Wissenschaften, Vol. 194, Springer-Verlag, New York–Heidelberg, 1972.

    Book  MATH  Google Scholar 

  13. J.-I. Igusa, On Jacobi’s derivative formula and its generalizations, American Journal of Mathematics 102 (1980), 409–446.

    Article  MathSciNet  MATH  Google Scholar 

  14. J.-I. Igusa, On the Nullwerte of Jacobians of odd theta functions, in Symposia Mathematica, Vol. XXIV, (Sympos., INDAM, Rome 1979), Academic Press, London–New York, 1981, pp. 83–95.

    Google Scholar 

  15. J.-I. Igusa, Multiplicity one theorem and problems related to Jacobi’ s formula, American Journal of Mathematics 105 (1983), 157–187.

    Article  MathSciNet  MATH  Google Scholar 

  16. D. Lehavi, Any smooth plane quartic can be reconstructed from its bitangents, Israel Journal of Mathematics 146 (2005), 371–379.

    Article  MathSciNet  MATH  Google Scholar 

  17. D. Plaumann, B. Sturmfels and C. Vinzant, Quartic curves and their bitangents, Journal of Symbolic Computation 46 (2011), 712–733.

    Article  MathSciNet  MATH  Google Scholar 

  18. H. Rauch and H. Farkas, Theta functions with applications to Riemann surfaces, The Williams&Wilkins Co., Baltimore, MD, 1974.

    MATH  Google Scholar 

  19. G. Salmon, A Treatise on the Higher Plane Curves: Intended as a Sequel to “Treatise on Conic Sections”, Chelsea, New York, 1960.

    MATH  Google Scholar 

  20. R. Salvati Manni, On the nonidentically zero Nullwerte of Jacobians of theta functions with odd characteristics, Advances in Mathematics 47 (1983), 88–104.

    Article  MathSciNet  MATH  Google Scholar 

  21. R. Salvati Manni, On the dimension of the vector space C[θ m]4. Nagoya Mathematical Journal 98 (1985), 99–107.

    Article  MathSciNet  MATH  Google Scholar 

  22. R. Salvati Manni, Modular varieties with level 2 theta structure, American Journal of Mathematics 116 (1994), 1489–1511.

    Article  MathSciNet  MATH  Google Scholar 

  23. C. Swierczewski et al., abelfunctions: A library for computing with Abelian functions, Riemann surfaces, and algebraic curves, http://abelfunctions.cswiercz.info, 2015.

    Google Scholar 

  24. C. Swierczewski and B. Deconinck, Computing Riemann theta functions in Sage with applications, Mathematics and Computers in Simulation 127 (2016), 263–272.

    Article  MathSciNet  Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Dipartimento di Matematica, Università “La Sapienza”, Piazzale A. Moro 2, I-00185, Roma, Italy

    Francesco Dalla Piazza, Alessio Fiorentino & Riccardo Salvati Manni

Authors
  1. Francesco Dalla Piazza
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Alessio Fiorentino
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Riccardo Salvati Manni
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Riccardo Salvati Manni.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Dalla Piazza, F., Fiorentino, A. & Manni, R.S. Plane quartics: the universal matrix of bitangents. Isr. J. Math. 217, 111–138 (2017). https://doi.org/10.1007/s11856-017-1440-z

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11856-017-1440-z

Search

Navigation