Some New Upper Bounds for the Size of Partial Ovoids in Slim Generalized Polygons and Generalized Hexagon of Order (s, s3)
Kris Coolsaet
and Hendrik Van Maldeghem
DOI: 10.1023/A:1026579608364
Abstract
From an elementary observation, we derive some upper bounds for the number of mutually opposite points in the classical generalized polygons having 3 points on each line. In particular, it follows that the Ree-Tits generalized octagon O(2) of order (2, 4) has no ovoids. Also, we deduce from another observation a similar upper bound in any generalized hexagon of order ( s, s 3).
Pages: 107–113
Keywords: generalized polygon; generalized hexagon; ovoid; partial ovoid; projective embedding
Full Text: PDF
References
1. A.E. Brouwer, A.M. Cohen, and A. Neumaier, “Distance-regular graphs,” Ergeb. Math. Grenzgeb. 18(3), (1989). Springer-Verlag, Berlin.
2. A.M. Cohen, “Point-line geometries related to buildings,” in Handbook of Incidence Geometry, Buildings and Foundations, F. Buekenhout (Ed.), North-Holland, Amsterdam, Chapter 9, 1995, pp. 647-737.
3. A.M. Cohen and J. Tits, “On generalized hexagons and a near octagon whose lines have three points,” European J. Combin. 6 (1985), 13-27.
4. W. Feit and G. Higman, “The nonexistence of certain generalized polygons,” J. Algebra 1 (1964), 114-131.
5. D.E. Frohardt and S.D. Smith, “Universal embeddings for the 3 D4(2) hexagon and the J2 near-octagon,” European J. Combin. 13 (1992), 455-472.
6. S.E. Payne and J.A. Thas, Finite Generalized Quadrangles, Pitman, Boston, 1984.
7. J.A. Thas, “Ovoids and spreads of finite classical polar spaces,” Geom. Dedicata 10 (1981), 135-144.
8. J. Tits, “Sur la trialité et certains groupes qui s'en déduisent,” Inst. Hautes Études Sci. Publ. Math. 2 (1959), 13-60.
9. H. Van Maldeghem, Generalized Polygons, Birkh\ddot auser Verlag, Basel,
1998. Monographs in Mathematics, Vol. 93.
2. A.M. Cohen, “Point-line geometries related to buildings,” in Handbook of Incidence Geometry, Buildings and Foundations, F. Buekenhout (Ed.), North-Holland, Amsterdam, Chapter 9, 1995, pp. 647-737.
3. A.M. Cohen and J. Tits, “On generalized hexagons and a near octagon whose lines have three points,” European J. Combin. 6 (1985), 13-27.
4. W. Feit and G. Higman, “The nonexistence of certain generalized polygons,” J. Algebra 1 (1964), 114-131.
5. D.E. Frohardt and S.D. Smith, “Universal embeddings for the 3 D4(2) hexagon and the J2 near-octagon,” European J. Combin. 13 (1992), 455-472.
6. S.E. Payne and J.A. Thas, Finite Generalized Quadrangles, Pitman, Boston, 1984.
7. J.A. Thas, “Ovoids and spreads of finite classical polar spaces,” Geom. Dedicata 10 (1981), 135-144.
8. J. Tits, “Sur la trialité et certains groupes qui s'en déduisent,” Inst. Hautes Études Sci. Publ. Math. 2 (1959), 13-60.
9. H. Van Maldeghem, Generalized Polygons, Birkh\ddot auser Verlag, Basel,
1998. Monographs in Mathematics, Vol. 93.