Flocks of cones of higher degree
Peter Sziklai
Eötvös University Department of Computer Science Budapest Pázmány P. s. 1/c Budapest H-1117 Hungary
DOI: 10.1007/s10801-006-0036-1
Abstract
It is known that in PG(3, q), q > 19, a partial flock of a quadratic cone with q-ϵ planes, can be extended to a unique flock if e < 1/ 4 Ö q ε<{1\over 4}\sqrt{q}, and a similar and slightly stronger theorem holds for the case q even. In this paper we prove the analogue of this result for cones with base curve of higher degree.
Pages: 233–238
Keywords: keywords flock; cone
Full Text: PDF
References
1. L. Bader, Flocks of cones and related structures, lecture notes.
2. R.D. Baker, G.L. Ebert, and T. Penttila, “Hyperbolic fibrations and q-clans,” Des. Codes Crypt. 34 (2005), 295-305.
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8. W.M. Kantor, “Note on generalized quadrangles, flocks and BLT-sets,” J. Comb. Theory Ser. (A) 58 (1991), 153-157.
9. P. Sziklai, “On subsets of GF(q2) with d-th power differences,” Discrete Math. 208-209 (1999), 547- 555.
10. P. Sziklai, “Partial flocks of the quadratic cone,” J. Combin. Th. Ser. A 113 (2006), 698-702.
11. L. Storme, and J. A. Thas, “k-arcs and partial flocks,” Linear Algebra Appl. 226/228 (1995), 33-45.
12. T. Sz\Acute\Acute onyi, “On the number of directions determined by a set of points in an affine Galois plane,” J. Combin. Th. Ser. A 74 (1996), 141-146.
13. J.A. Thas, “Generalized Quadrangles and Flock of Cones,” Europ. J. Combin. 8 (1987), 441-451.
14. M. Walker, “A class of translation planes,” Geom. Dedicata 5 (1976), 135-146.
15. A. Weil, “Sur les Courbes Algébrique et les varietés qui s'en déduisent,” Actualités Scientifiques et Industrielles 1041, Herman & Cie, Paris, 1948.
2. R.D. Baker, G.L. Ebert, and T. Penttila, “Hyperbolic fibrations and q-clans,” Des. Codes Crypt. 34 (2005), 295-305.
3. L. Bader, G. Lunardon, and J.A. Thas, “Derivation of flocks of quadratic cones,” Forum Math. 2 (1990), 163-174.
4. J.C. Fisher and J.A. Thas, “Flocks in PG(3, q),” Math. Z. 169 (1979), 1-11.
5. H. Gevaert and N.L. Johnson, “Flocks of quadratic cones, generalised quadrangles and translation planes,” Geom. Dedicata 27 (1988), 301-317.
6. J.W.P. Hirschfeld, Projective geometries over finite fields, 2nd edition, Oxford University Press, 1998.
7. N. Knarr, “A geometric construction of generalized quadrangles from polar spaces of rank three,” Resultate Math. 21 (1992), 332-344.
8. W.M. Kantor, “Note on generalized quadrangles, flocks and BLT-sets,” J. Comb. Theory Ser. (A) 58 (1991), 153-157.
9. P. Sziklai, “On subsets of GF(q2) with d-th power differences,” Discrete Math. 208-209 (1999), 547- 555.
10. P. Sziklai, “Partial flocks of the quadratic cone,” J. Combin. Th. Ser. A 113 (2006), 698-702.
11. L. Storme, and J. A. Thas, “k-arcs and partial flocks,” Linear Algebra Appl. 226/228 (1995), 33-45.
12. T. Sz\Acute\Acute onyi, “On the number of directions determined by a set of points in an affine Galois plane,” J. Combin. Th. Ser. A 74 (1996), 141-146.
13. J.A. Thas, “Generalized Quadrangles and Flock of Cones,” Europ. J. Combin. 8 (1987), 441-451.
14. M. Walker, “A class of translation planes,” Geom. Dedicata 5 (1976), 135-146.
15. A. Weil, “Sur les Courbes Algébrique et les varietés qui s'en déduisent,” Actualités Scientifiques et Industrielles 1041, Herman & Cie, Paris, 1948.