Geometriae Dedicata

, Volume 150, Issue 1, pp 151–180 | Cite as

Ruled quartic surfaces, models and classification

  • Irene Polo-Blanco
  • Marius van der Put
  • Jaap Top
Open Access
Original Paper


New historical aspects of the classification, by Cayley and Cremona, of ruled quartic surfaces and the relation to string models and plaster models are presented. In a ‘modern’ treatment of the classification of ruled quartic surfaces the classical one is corrected and completed. The string models of Series XIII of some ruled quartic surfaces (manufactured by L. Brill and by M. Schilling) are based on a result of Rohn concerning curves in \({\mathbb{P}^1\times \mathbb{P}^1}\) of bi-degree (2, 2). This is given here a conceptional proof.


Ruled surface Quartic suface Grassmann variety Dual surface Reciprocal surface 

Mathematics Subject Classification (2000)

14-03 01-02 14J26 14M15 14N25 32S25 



The authors thank the referee for his very useful remarks.

Open Access

This article is distributed under the terms of the Creative Commons Attribution Noncommercial License which permits any noncommercial use, distribution, and reproduction in any medium, provided the original author(s) and source are credited.


  1. 1.
    Barth W., Peters C., van der Ven A.: Compact complex surfaces. Springer, Berlin (1984)MATHGoogle Scholar
  2. 2.
    Bottema O.: A classification of rational quartic ruled surfaces. Geometriae Dedicata 1(3), 349–355 (1973)MathSciNetMATHCrossRefGoogle Scholar
  3. 3.
    Cayley A.: A third memoir on skew surfaces, otherwise scrolls. Philos. Trans. R. Soc. Lond. 159, 111–126 (1869)MATHCrossRefGoogle Scholar
  4. 4.
    Chasles M.: Sur les six droites qui peuvent être les directions de six forces en équilibre. Propriétés de l’hyperboloïde à une nappe et d’une certaine surface du quatrième ordre. Comptes Rendus des Séances de l’Académie des Sciences. Paris 52, 1094–1104 (1861)Google Scholar
  5. 5.
    Cremona, L.: Sulle Superficie Gobbe di Quarto Grado. Memorie dell’ Accademia delle Science dell’ Istituto di Bologna serie II, tomo VIII (1868), 235–250. Opere II, 420Google Scholar
  6. 6.
    Dolgachev, I.V.: Topics in Classical Algebraic Geometry,
  7. 7.
    Edge W.L.: The theory of ruled surfaces. Cambridge University Press, Cambridge (1931)Google Scholar
  8. 8.
    Hartshorne R.: Algebraic geometry. Springer, Berlin (1977)MATHGoogle Scholar
  9. 9.
    Meyer W.: Fr. Flächen vierter und höherer Ordnung. Encyklopädie der mathematischen Wissenschaften mit Einschluss ihrer Anwendungen IIIC 10b, 1744–1759 (1930)Google Scholar
  10. 10.
    Mohrmann H.: Die Flächen vierter Ordnung mit gewundener Doppelkurve. Mathematische Annalen 89, 1–31 (1923)MathSciNetMATHCrossRefGoogle Scholar
  11. 11.
    Pascal, E.: Repertorium der Höheren Mathematik (Definitionen, Formeln, Theoreme, Literatur). II. Theil: die Geometrie. Teubner (1902)Google Scholar
  12. 12.
    Rohn K.: Ueber die Flächen vierter Ordnung mit dreifachem Punkte. Mathematische Annalen 24(1), 55–151 (1884)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Rohn, K.: Die verschiedenen Arten der Regelflächen vierter Ordnung, (1886). Mathematische Abhandlungen aus dem Verlage Mathematischer Modelle von Martin Schilling. Halle a. S., 1904. Mathematische Annalen, vol. 28, No. 2, pp. 284–308 (1886)Google Scholar
  14. 14.
    Salmon G.: A treatise on the analytic geometry of three dimensions. Hodges, Figgis & Co., Dublin (1882)Google Scholar
  15. 15.
    Schilling, M.: Catalog mathematischer Modelle für den höheren mathematischen Unterrich. Verlag von Martin Schilling, Leipzig (1911)Google Scholar
  16. 16.
    Segre C.: Etude des différentes surfaces du 4e ordre à conique double ou cuspidale (générale ou décomposée) considérées comme des projections de l’intersection de deux variétés quadratiques de l’espace à quatre dimensions. Mathematische Annalen 24(3), 313–444 (1884)MathSciNetCrossRefGoogle Scholar
  17. 17.
    Sturm R.: Die Gebilde ersten und zweiten Grades der Liniengeometrie in synthetischer Behandlung I. B.G. Teubner, Leipzig (1892)MATHGoogle Scholar
  18. 18.
    Swinnerton-Dyer H.P.F.: An enumeration of all varieties of degree 4. Am J Math 95(2), 403–418 (1973)MathSciNetMATHCrossRefGoogle Scholar
  19. 19.
    Urabe T.: Dynkin graphs and combinations of singularities on quartic surfaces. Proc. Japan. Acad. Ser. A 61, 266–269 (1985)MathSciNetMATHCrossRefGoogle Scholar
  20. 20.
    Urabe T.: Classification of non-normal quartic surfaces. Tokyo J Math 9(2), 265–295 (1986)MathSciNetMATHCrossRefGoogle Scholar
  21. 21.
    Urabe T.: Elementary transformations of Dynkin graphs and singularities on quartic surfaces. Inventiones mathematicae 87, 549–572 (1987)MathSciNetMATHCrossRefGoogle Scholar
  22. 22.
    Urabe T.: Tie transformations of Dynkin graphs and singularities on quartic surfaces. Inventiones mathematicae 100, 207–230 (1990)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Wong B.C.: A study and classifications of ruled quartic surfaces by means of a point-to-line transformation. Univ. Calif. Publ. Math. 1(17), 371–387 (1923)Google Scholar

Copyright information

© The Author(s) 2010

Authors and Affiliations

  • Irene Polo-Blanco
    • 1
  • Marius van der Put
    • 2
  • Jaap Top
    • 2
  1. 1.Department MatescoUniversity of CantabriaSantaderSpain
  2. 2.Department of MathematicsUniversity of GroningenGroningenThe Netherlands

Personalised recommendations