Hostname: page-component-78c5997874-s2hrs Total loading time: 0 Render date: 2024-11-10T20:14:53.829Z Has data issue: false hasContentIssue false
  • English
  • Français

Explicit Real Cubic Surfaces

Published online by Cambridge University Press:  20 November 2018

Irene Polo-Blanco  and
Jaap Top
Irene Polo-Blanco
Affiliation:
IwI-RuG, P.O. Box 800, 9700 AV Groningen, The Netherlands e-mail: j.top@rug.nl
Jaap Top
Affiliation:
IwI-RuG, P.O. Box 800, 9700 AV Groningen, The Netherlands e-mail: j.top@rug.nl
Rights & Permissions [Opens in a new window]

Abstract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

The topological classification of smooth real cubic surfaces is recalled and compared to the classification in terms of the number of real lines and of real tritangent planes, as obtained by $\text{L}$. Schläfli in 1858. Using this, explicit examples of surfaces of every possible type are given.

Keywords

14J2514J8014P2514Q10
Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 51 , Issue 1 , 01 March 2008 , pp. 125 - 133
Copyright
Copyright © Canadian Mathematical Society 2008

References

[AKP] Artebani, A., Kloosterman, R., and Pacini, M., A new model for the theta divisor of the cubic threefold. Le Matematiche, 58(2003), no. 2, 201–236.Google Scholar
[B] Beauville, A., Complex Algebraic Surfaces. London Mathematical Society Lecture Note Series 68. Cambridge University Press, Cambridge, 1983.Google Scholar
[Cl] Clebsch, A., Ueber die Anwendung der quadratischen Substitution auf die Gleichungen 5ten Grades und die geometrische Theorie des ebenen Fünfseits. Math. Ann. 4(1871), no. 2, 284–345.Google Scholar
[Cr] Cremona, L., Mémoire de géométrie pure sur les surfaces du troisiéme ordre. J. Math Pures Appl. 68(1868), 1–133.Google Scholar
[El] Elkies, N., Complete cubic parametrization of the Fermat cubic surface w 3 + x 3 + y 3 + z 3 = 0. http://www.math.harvard.edu/_elkies/4cubes.html Google Scholar
[F] Fischer, G., ed. Mathematische Modelle, Vieweg, Braunschweig, 1986.Google Scholar
[Ha] Hartshorne, R., Algebraic Geometry. Graduate Texts in Mathematics 52. Springer-Verlag, New York, 1977.Google Scholar
[HL] Holzer, S., and Labs, O., Illustrating the classification of real cubic surfaces. In: Algebraic Geometry and Geometric Modeling. Springer, Berlin, 2006, pp. 119–134.Google Scholar
[Hu] Hunt, B., The Geometry of some special Arithmetic Quotients, Lecture Notes in Mathematics 1637, Springer-Verlag, Berlin, 1996.Google Scholar
[KM] Knörrer, H., and Miller, T., Topologische Typen reeller kubischer Flächen. Math. Z. 195(1987), no. 1, 51–67.Google Scholar
[K] Kollár, J., Real algebraic surfaces, arXiv:AG/9712003, 1997.Google Scholar
[M] Manin, Y. I., Cubic Forms. Second edition. North-Holland, 1974, 1986.Google Scholar
[Po1] Polo-Blanco, I., Mathematical models of surfaces. Inventory, Univ. of Groningen, 2004 http://www.math.rug.nl/models Google Scholar
[Po2] Polo-Blanco, I., Theory and History of Geometric Models. Ph.D. thesis, Groningen, 2007. http://dissertations.ub.rug.nl/faculties/science/2007/i.polo.blanco/.Google Scholar
[Sch] Schläfli, L., An attempt to determine the twenty-seven lines upon a surface of the third order, and to divide such surfaces into species in reference to the reality of the lines upon the surface, Quart. J. Pure Appl. Math. 2 (1858), 110–120.Google Scholar
[Se] Segre, B., The Non-Singular Cubic Surfaces. Oxford, Oxford University Press, 1942.Google Scholar
[Sh] Shioda, T., Weierstrass transformations and cubic surfaces. Comment. Math. Univ. St. Paul. 44(1995), no. 1, 109–128.Google Scholar
[Si] Silhol, R., Real Algebraic Surfaces, Lecture Notes in Mathematics 1392, Springer-Verlag, Berlin, 1989.Google Scholar