Designs, Codes and Cryptography

, Volume 87, Issue 4, pp 931–953 | Cite as

Cubic surfaces over small finite fields

  • Anton BettenEmail author
  • Fatma Karaoglu
Part of the following topical collections:
  1. Special Issue: Finite Geometries


In the 1960s, Hirschfeld embarked on a program to classify cubic surfaces with 27 lines over finite fields. This work is a contribution to this problem. We develop an algorithm to classify surfaces with 27 lines over a finite field using the classical theory of double-sixes. This algorithm is used to classify these surfaces over all fields of order q at most 97. We then construct a family of cubic surfaces over finite fields of odd order. The generic surfaces in this family have six Eckardt points and they are invariant under a symmetric group of degree four. The family turns out to be isomorphic to the example of a family of cubic surface given over the real numbers by Hilbert and Cohn-Vossen.


Geometry Cubic surface Finite field Classification Double-six 

Mathematics Subject Classification

05-04 05B25 14J10 51E25 68R05 



Both authors thank James Hirschfeld for helpful conversations. The first author thanks Steve Linton for helpful remarks about the action of the general linear group on the wedge product. Finally, we thank all three reviewers of this paper for the helpful comments. The paper has been improved greatly because of these comments.


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Authors and Affiliations

  1. 1.Department of MathematicsColorado State UniversityFort CollinsUSA
  2. 2.Department of MathematicsUniversity of SussexBrightonUK

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