A Note on k-Secant Lines of the Klein Cubic Threefold in PG(4,3)

Yıl 2025, Cilt: 18 Sayı: 3, 703 - 712, 31.12.2025

Ayşe Bayar , Ziya Akça

https://izlik.org/JA75XA84MU

Öz

In this paper, we developed an algorithm to classify and construct k-secant lines for the Klein cubic threefold F in the projective space PG(4,3). The algorithm identifies secant lines based on their intersections with F, revealing distinct categories of non-secant and k-secant lines. Additionally, we partitioned the point set of F into three subsets, uncovering geometric configurations and forming 5-gons related to perspectively. This analysis offers new insights into the structure of the Klein cubic threefold and its geometric properties.

Anahtar Kelimeler

Klein cubic threefold , Projective spaces , Galois field , secant line.

Kaynakça

• [1] Akça, Z., Bayar, A., Ekmekçi, S., Kaya, R., Thas, J A., Van Maldeghem, H., (2012) Generalized Veronesean embeddings of projective spaces Part II The lax case, Ars Combinatoria, vol. 103, 65-80.
• [2] Akça, Z., Bayar, A., Ekmekçi, S., Kaya, R., Van Maldeghem, H., (20120, Bazı geometrilerin sonlu projektif uzaylara gömülmeleri üzerine, Tübitak Proje No: 108T340.
• [3] Altıntaş, A., Bayar, A., Ekmekçi, S., Akça, Z.,, (2024) On the Klein cubic threefold in PG(4,2), International J.Math. Combin. Vol.4(2024), 9-19.
• [4] Bayar, A., Akça, Z., Ekmekçi, S., (2022) On embedding the projective plane PG(2,4) to the projective space PG(4,4), New Trends in Mathematical Sciences, 10(4), 142-150.
• [5] Betten, A., Karaoglu, F., (2019) Cubic surfaces over small finite fields. Designs, Codes and Cryptography, 87 (4): 931-953.
• [6] Dickson, L.E., (1915) Projective classification of cubic surfaces modulo 2. Annal sof Mathematics,16: 139-157.
• [7] Ekmekçi, S., Bayar, A., Akça, Z., (2022) On the projective planes in projective space PG(4,4), Erciyes Üniversitesi Fen Bilimleri Enstitüsü Fen Bilimleri Dergisi, 38(3), 519–524.
• 8] Hirscfeld, JWP., (1967) Classical configurations over finite fields. I. The doublesix and the cubic surface with 27 lines. Rendiconti di Matematica e delle sue Applicazioni, 26 (5): 115- 152.
• [9] Karaoglu, F., Nonsingular cubic surfaces over2kF , (2021) Turkish Journal of Mathematics, 45: 2492-2510.
• [10] Karaoglu, F., (2022) Smooth cubic surfaces with 15 lines, Applicable Algebra in Engineering, Communication and Computing, 33(16) 823-853.
• [11] Klein, F.,. (1879) Uber die Ausásung Gleichungen siebenten und achten Grades. Math. Ann., 15, 251-282.
• [12] Rosati, L.A., (1957) L’equazione delle 27 rette della superficie cubica generale in un corpo fiito. Bollettino dell’Unione Matematica Italiana, 12(3): 612-626.
• [13] Bockondas, G., Bossoto, B., Eckart points on a cubic threefold, https://arxiv.org/pdf/2309.08124. (13.11.2024).