The number of cubic surfaces with 27 lines over a finite field

Abstract

We determine the number of cubic surfaces with 27 lines over a finite field \({{\mathbb {F}}}_q\). This is based on exploiting the relationship between non-conical six-arcs in a projective plane embedded in projective three-space and cubic surfaces with 27 lines. We revisit this classical relationship, which goes back to work of Clebsch in the nineteenth century. Our result can be used as an enumerative check for a computer classification of cubic surfaces with 27 lines over finite fields.

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Acknowledgements

The authors thank the two referees for helpful comments which have improved the overall presentation.

Funding

Funding was provided by The Scientific and Technological Research Council of Turkey (Grant Nos. 1059B211900062 and 1059B192000479).

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Correspondence to Fatma Karaoglu.

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Karaoglu, F., Betten, A. The number of cubic surfaces with 27 lines over a finite field. J Algebr Comb (2021). https://doi.org/10.1007/s10801-020-01009-3

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Keywords

  • Geometry
  • Cubic surface
  • Finite field
  • Counting

Mathematics Subject Classification

  • 05B25
  • 05E18
  • 14E05
  • 14J26
  • 51E25