Abstract
Given an orthogonal-Buekenhout–Metz unital Uα,β, embedded in PG(2, q2), and a point P ∉ Uα,β, we study the set τP(Uα,β) of feet of P in Uα,β. We characterize geometrically each of these sets as either q + 1 collinear points or as q + 1 points partitioned into two arcs. Other results about the geometry of these sets are also given.
Acknowledgements
The authors would like to thank Universidad de Chile, and its Stimulus Program for Institutional Excellence for supporting the third author’s visit to Universidad de Chile, where a part of this work was done. Also, the third author would like to thank the California State University Chancellor’s office’s research, scholarship& creative activities award for its support during Fall 2015.
Funding: During the time this project was done, the second author was funded by Fondecyt project # 1140510.
References
[1] A. Aguglia, G. L. Ebert, A combinatorial characterization of classical unitals. Arch. Math. (Basel)78 (2002), 166–172. MR1888419 Zbl 1006.5100410.1007/s00013-002-8231-3Search in Google Scholar
[2] R. D. Baker, G. L. Ebert, Intersection of unitals in the Desarguesian plane. In: Proceedings of the Twentieth Southeastern Conference on Combinatorics, Graph Theory, and Computing (Boca Raton, FL, 1989), volume 70, 87–94, 1990. MR1041587 Zbl 0695.51009Search in Google Scholar
[3] S. Barwick, G. Ebert, Unitals in projective planes. Springer 2008. MR2440325 Zbl 1156.51006Search in Google Scholar
[4] F. Buekenhout, Existence of unitals in finite translation planes of order q2 with a kernel of order q. Geometriae Dedicata5 (1976), 189–194. MR0448236 Zbl 0336.5001410.1007/BF00145956Search in Google Scholar
[5] P. Dembowski, Finite geometries. Springer 1997. MR1434062 Zbl 0865.51004Search in Google Scholar
[6] N. Durante, A. Siciliano, Unitals of PG(2, q2) containing conics. J. Combin. Des. 21 (2013), 101–111. MR3011984 Zbl 1273.0502510.1002/jcd.21314Search in Google Scholar
[7] J. W. P. Hirschfeld, Projective geometries over finite fields. Oxford Univ. Press 1998. MR1612570 Zbl 0899.51002Search in Google Scholar
[8] J. W. P. Hirschfeld, T. Szőnyi, Sets in a finite plane with few intersection numbers and a distinguished point. Discrete Math. 97 (1991), 229–242. MR1140805 Zbl 0748.5101110.1016/0012-365X(91)90439-9Search in Google Scholar
[9] V. Krčadinac, K. Smoljak, Pedal sets of unitals in projective planes of order 9 and 16. Sarajevo J. Math. 7(20) (2011), 255–264. MR2906536 Zbl 1277.51010Search in Google Scholar
[10] J. A. Thas, A combinatorial characterization of Hermitian curves. J. Algebraic Combin. 1 (1992), 97–102. MR1162643 Zbl 0784.5102310.1023/A:1022437415099Search in Google Scholar
© 2018 Walter de Gruyter GmbH Berlin/Boston