Smooth cubic surfaces with 15 lines

Karaoglu F.

Applicable Algebra in Engineering, Communications and Computing, cilt.33, sa.6, ss.823-853, 2022 (SCI-Expanded) identifier identifier

  • Yayın Türü: Makale / Tam Makale
  • Cilt numarası: 33 Sayı: 6
  • Basım Tarihi: 2022
  • Doi Numarası: 10.1007/s00200-022-00582-3
  • Dergi Adı: Applicable Algebra in Engineering, Communications and Computing
  • Derginin Tarandığı İndeksler: Science Citation Index Expanded (SCI-EXPANDED), Scopus, Academic Search Premier, Applied Science & Technology Source, Computer & Applied Sciences, INSPEC, MathSciNet, zbMATH
  • Sayfa Sayıları: ss.823-853
  • Anahtar Kelimeler: Non-algebraically closed fields, Cubic surfaces, Classification
  • Kocaeli Üniversitesi Adresli: Hayır

Özet

© 2022, The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.In this paper, we consider smooth cubic surfaces with 15 lines. It is known that such surfaces can be generated by means of a double six with two pairs of Galois conjugate lines defined over the quadratic extension. The approach taken here is to consider the generation by means of a set of 9 lines defined over the field of coordinates. Eight lines arise from the double six, while the ninth is the diagonal line of the two pairs of Galois conjugate lines. This allows us to express all necessary equations and objects in terms of a set of four parameters over the coordinate field. As an application, we classify the smooth cubic surfaces with 15 lines over small finite fields by computer. All our results match with an enumerative formula recently found by Das.