Reconstructing Plane Quartics from Their Invariants

  • Reynald Lercier
  • Christophe Ritzenthaler
  • Jeroen SijslingEmail author


We present an explicit method that, given a generic tuple of Dixmier–Ohno invariants, reconstructs a corresponding plane quartic curve.


Plane quartic curves Invariant theory Dixmier–Ohno invariants Moduli spaces Reconstruction 

Mathematics Subject Classification

13A50 14L24 14H10 14H25 



We would like to thank the participants of the working group TEDI, and in particular Boris Kolev and Marc Olive, for their interest and for the many useful discussions.


  1. 1.
    Artebani, M., Quispe, S.: Fields of moduli and fields of definition of odd signature curves. Arch. Math. (Basel) 99(4), 333–344 (2012)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Basson, R.: Arithmétique des espaces de modules des courbes hyperelliptiques de genre \(3\) en caractéristique positive. PhD thesis, Université de Rennes 1 (2015).
  3. 3.
    Böhning, C.: The rationality of the moduli space of curves of genus 3 after P. Katsylo. In: Bogomolov, F., Tschinkel, Yu., (eds.) Cohomological and Geometric Approaches to Rationality Problems. Progress in Mathematics, vol. 282, pp. 17–53. Birkhäuser, Boston (2010)Google Scholar
  4. 4.
    Bosma, W., Cannon, J., Playoust, C.: The Magma algebra system. I. The user language. J. Symb. Comput. 24(3–4), 235–265 (1997)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Brouwer, A.E., Popoviciu, M.: The invariants of the binary decimic. J. Symb. Comput. 45(8), 837–843 (2010)MathSciNetCrossRefGoogle Scholar
  6. 6.
    Brouwer, A.E., Popoviciu, M.: The invariants of the binary nonic. J. Symb. Comput. 45(6), 709–720 (2010)MathSciNetCrossRefGoogle Scholar
  7. 7.
    Clebsch, A.: Theorie der binären algebraischen Formen. B.G. Teubner, Leipzig (1872)zbMATHGoogle Scholar
  8. 8.
    Cohen, H., Frey, G., Avanzi, R., Doche, C., Lange, T., Nguyen, K., Vercauteren, F. (eds.): Handbook of Elliptic and Hyperelliptic Curve Cryptography. Discrete Mathematics and its Applications (Boca Raton). Chapman & Hall/CRC, Boca Raton (2006)Google Scholar
  9. 9.
    Dixmier, J.: On the projective invariants of quartic plane curves. Adv. Math. 64(3), 279–304 (1987)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Dolgachev, I.: Lectures on Invariant Theory. London Mathematical Society Lecture Note Series, 296th edn. Cambridge University Press, Cambridge (2003)CrossRefGoogle Scholar
  11. 11.
    Elsenhans, A.-S.: Explicit computations of invariants of plane quartic curves. J. Symb. Comput. 68(2), 109–115 (2015)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Freiherr von Gall, A.: Das vollständige Formensystem einer binären Form achter Ordnung. Math. Ann. 17(1), 31–51, 139–152, 456 (1880)Google Scholar
  13. 13.
    Freiherr von Gall, A.: Das vollständige Formensystem der binären Form 7ter Ordnung. Math. Ann. 31(3), 318–336 (1888)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Fulton, W., Harris, J.: Representation Theory. Graduate Texts in Mathematics, vol. 129. Springer, New York (1991)zbMATHGoogle Scholar
  15. 15.
    Gatti, V., Viniberghi, E.: Spinors of \(13\)-dimensional space. Adv. Math. 30(2), 137–155 (1978)MathSciNetCrossRefGoogle Scholar
  16. 16.
    Girard, M., Kohel, D.R.: Classification of genus 3 curves in special strata of the moduli space. In: Hess, F., Pauli, S., Pohst, M. (eds.) Algorithmic Number Theory. Lecture Notes in Computer Science, vol. 4076, pp. 346–360. Springer, Berlin (2006)CrossRefGoogle Scholar
  17. 17.
    Gordan, P.: Beweis, dass jede Covariante und Invariante einer binären Form eine ganze Function mit numerischen Coefficienten einer endlichen Anzahl solcher Formen ist. J. Reine Angew. Math. 69, 323–354 (1868)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Hashimoto, M.: Equivariant total ring of fractions and factoriality of rings generated by semi-invariants. Commun. Algebra 43(4), 1524–1562 (2015)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Hilbert, D.: Theory of Algebraic Invariants. Cambridge University Press, Cambridge: Translated from the German and with a preface by Reinhard C. Laubenbacher, Edited and with an introduction by Bernd Sturmfels (1993)Google Scholar
  20. 20.
    Katsylo, P.I.: On the birational geometry of the space of ternary quartics. In: Vinberg, E.B. (ed.) Lie Groups, Their Discrete Subgroups, and Invariant Theory. Advances in Soviet Mathematics, vol. 8, pp. 95–103. American Mathematical Society, Providence (1992)CrossRefGoogle Scholar
  21. 21.
    Katsylo, P.: Rationality of the moduli variety of curves of genus 3. Comment. Math. Helv. 71(4), 507–524 (1996)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Kılıçer, P., Labrande, H., Lercier, R., Ritzenthaler, C., Sijsling, J., Streng, M.: Plane quartics over \({\mathbb{Q}}\) with complex multiplication. Acta Arith. 185(2), 127–156 (2018). arXiv:1701.06489 MathSciNetCrossRefGoogle Scholar
  23. 23.
    Kraft, H., Procesi, C.: Classical Invariant Theory. A Primer (1996). Notes available at
  24. 24.
    Lercier, R., Lorenzo García, E., Ritzenthaler, C.: Reduction type of non-hyperelliptic genus 3 curves (2018). arXiv:1803.05816
  25. 25.
    Lercier, R., Ritzenthaler, C.: Hyperelliptic curves and their invariants: geometric, arithmetic and algorithmic aspects. J. Algebra 372, 595–636 (2012)MathSciNetCrossRefGoogle Scholar
  26. 26.
    Lercier, R., Ritzenthaler, C., Sijsling, J.: Fast computation of isomorphisms of hyperelliptic curves and explicit Galois descent. In: Howe, E.W., Kedlaya, K.S. (eds.) Proceedings of the Tenth Algorithmic Number Theory Symposium, pp. 463–486. Mathematical Sciences Publishers, Berkeley (2012)MathSciNetCrossRefGoogle Scholar
  27. 27.
    Lercier, R., Ritzenthaler, C., Rovetta, F., Sijsling, J.: Parametrizing the moduli space of curves and applications to smooth plane quartics over finite fields. LMS J. Comput. Math. 17(Suppl. A), 128–147 (2014)MathSciNetCrossRefGoogle Scholar
  28. 28.
    Lercier, R., Ritzenthaler, C., Sijsling, J.: quartic \(_{-}\) reconstruction: a Magma package for reconstructing plane quartics from Dixmier–Ohno invariants. (2016)
  29. 29.
    Looijenga, E.: Invariants of quartic plane curves as automorphic forms. In: Keum, J., Kondō, S. (eds.) Algebraic Geometry. Contemporary Mathematics, vol. 422, pp. 107–120. American Mathematical Society, Providence (2007)CrossRefGoogle Scholar
  30. 30.
    Lorenzo García, E.: Twists of non-hyperelliptic curves. Rev. Mat. Iberoam. 33(1), 169–182 (2017)MathSciNetCrossRefGoogle Scholar
  31. 31.
    Maeda, T.: On the invariant field of binary octavics. Hiroshima Math. J. 20(3), 619–632 (1990)MathSciNetzbMATHGoogle Scholar
  32. 32.
    Mestre, J.-F.: Construction de courbes de genre \(2\) à partir de leurs modules. In: Mora, T., Traverso, C. (eds.) Effective Methods in Algebraic Geometry. Progress in Mathematics, vol. 94, pp. 313–334. Birkhäuser, Boston (1991)CrossRefGoogle Scholar
  33. 33.
    Mumford, D., Fogarty, J., Kirwan, F.: Geometric Invariant Theory. Ergebnisse der Mathematik und ihrer Grenzgebiete, vol. 34, 3rd edn. Springer, Berlin (1994)CrossRefGoogle Scholar
  34. 34.
    Olive, M.: About Gordan’s algorithm for binary forms. Found. Comput. Math. 17(6), 1407–1466 (2017)MathSciNetCrossRefGoogle Scholar
  35. 35.
    Olver, P.J.: Classical Invariant Theory. London Mathematical Society Student Texts, vol. 44. Cambridge University Press, Cambridge (1999)Google Scholar
  36. 36.
    Popov, V.L.: Stability of the action of an algebraic group on an algebraic variety. Izv. Akad. Nauk SSSR Ser. Mat. 36, 371–385 (1972)MathSciNetzbMATHGoogle Scholar
  37. 37.
    Procesi, C.: Invariant Theory. Monografías del Instituto de Matemática y Ciencias Afines, vol. 2. Instituto de Matemática y Ciencias Afines, IMCA, Lima (1998)Google Scholar
  38. 38.
    Rökaeus, K.: Computer search for curves with many points among abelian covers of genus 2 curves. In: Aubry, Y., Ritzenthaler, C., Zykin, A. (eds.) Arithmetic, Geometry, Cryptography and Coding Theory. Contemporary Mathematics, vol. 574, pp. 145–150. American Mathematical Society, Providence (2012)CrossRefGoogle Scholar
  39. 39.
    Salmon, G.: A treatise on the higher plane curves: intended as a sequel to “A treatise on conic sections”, 3rd edn. Chelsea, New York (1960)zbMATHGoogle Scholar
  40. 40.
    Shioda, T.: On the graded ring of invariants of binary octavics. Am. J. Math. 89(4), 1022–1046 (1967)MathSciNetCrossRefGoogle Scholar
  41. 41.
    Springer, T.A.: Invariant Theory. Lecture Notes in Mathematics, vol. 585. Springer, Berlin (1977)Google Scholar
  42. 42.
    Stoll, M.: Reduction theory of point clusters in projective space. Groups Geom. Dyn. 5(2), 553–565 (2011)MathSciNetCrossRefGoogle Scholar
  43. 43.
    Sturmfels, B.: Algorithms in Invariant Theory. Texts and Monographs in Symbolic Computation, 2nd edn. Springer, Vienna (2008)Google Scholar
  44. 44.
    van Leeuwen, M.A.A., Cohen, A.M., Lisser, B.: LiE, A Package for Lie Group Computations. Computer Algebra Nederland, Amsterdam (1997)Google Scholar
  45. 45.
    van Rijnswou, S.M.: Testing the Equivalence of Planar Curves. PhD thesis, Technische Universiteit Eindhoven, Eindhoven (2001).
  46. 46.
    Weil, A.: The field of definition of a variety. Am. J. Math. 78, 509–524 (1956)MathSciNetCrossRefGoogle Scholar

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Authors and Affiliations

  1. 1.DGA & IRMARUniversité de Rennes 1RennesFrance
  2. 2.IRMARUniversité de Rennes 1RennesFrance
  3. 3.Department of MathematicsDartmouth CollegeHanoverUSA
  4. 4.Institut für Reine MathematikUniversität UlmUlmGermany

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