Flocks of Infinite Hyperbolic Quadrics
Norman L. Johnson
University of Iowa Mathematics Department Iowa City Iowa 52242
DOI: 10.1023/A:1008692117216
Abstract
Let K be a field containing a nonsquare F = K( Ö{ g} ) F = K\left( {\sqrt γ} \right) a quadratic extension. Let ( K( Ö{ g} ) * ) s+ 1 = K - \left( {K\left( {\sqrt γ} \right)^* } \right)^{σ+ 1} = K^ - , a construction is given which produces large numbers of infinite nearfield and non nearfield flocks of an infinite hyperbolic quadric in PG(3, K).
Pages: 27–51
Keywords: flock; quadric; bol translation plane
Full Text: PDF
References
1. L. Bader, “Some new examples of flocks of Q+(3, q),” Geom. Dedicata 27 (1988), 213-218.
2. L. Bader and G. Lunardon, “On the flocks of Q+(3, q),” Geom. Dedicata 29 (1989), 177-183. P1: rba Journal of Algebraic Combinatorics KL365-02-Johnson November 7, 1996 11:3 FLOCKS OF INFINITE HYPERBOLIC QUADRICS 51
3. R.D. Baker and G.L. Ebert, “A nonlinear flock in the Minkowski plane of order 11,” Congressus Num. 58 (1987), 75-81.
4. M. Biliotti and N.L. Johnson, “Bilinear flocks of quadratic cones,” J. Geom., to appear.
5. M. Biliotti and N.L. Johnson, “Variations on a theme of Dembowski,” Proceedings AMS Conference in Honor of J.G. Ostrom, Marcel Dekker, 1996 (to appear).
6. A. Bonisoli, “On the sharply 1-transitive subsets of PGL(2, pm ),” J. Geom. 31 (1988), 32-41.
7. R.P. Burn, “Bol quasifields and Pappus,” Theorem Math. Z. 105 (1968), 351-364.
8. F. De Clerck and H. Van Maldeghem, “Flocks of an infinite quadratic cone,” Bull. Belgian Math. Soc., Simon Stevin, 2 (1994), 399-415.
9. P. Dembowski, Finite Geometries, Berlin, Heidelberg, New York: Springer-Verlag, 1968.
10. J. Hanson and M.J. Kallaher, “Finite Bol quasifields are nearfields,” Utilitas Math. 37 (1990), 45-64.
11. V. Jha and N.L. Johnson, “Infinite flocks of quadratic cones,” J. Geom., to appear.
12. N.L. Johnson, “Flocks of hyperbolic quadrics and translation planes admitting affine homologies,” J. Geom. 34 (1989), 50-73.
13. M.J. Kallaher, “A note on Bol projective planes,” Arch. d. Math. 20 (1969), 329-333.
14. M.J. Kallaher, “A note on finite Bol quasifields,” Arch. d. Math. 23 (1972), 164-166.
15. W.F. Orr, “The Miquelian inversive plane I P(q) and the associated projective planes,” Ph.D. Thesis, Univ. Wisconsin, 1973.
16. R. Riesinger, “Faserungen, die aus Reguli mit einem gemeinsamen Geradenpaar zusammengesetzt sind,” J. Geom. 45 (1992), 137-157.
17. J.A. Thas, “Flocks of egglike inversive planes,” in Finite Geometric Structures and their Applications, A. Barlotti (Ed.), Rome, 1973, 189-191.
18. J.A. Thas, “Flocks of nonsingular ruled quadrics in PG(3, q),” Rend. Accad. Nat. Lincei 59 (1975), 83-85.
19. J.A. Thas, “Generalized quadrangles and flocks of cones,” Europ. J. Comb. 8 (1987), 441-452.
20. J.A. Thas, “Flocks, maximal exterior sets and inversive planes,” Contemp. Math. 111 (1990), 187-218.
21. J.A. Thas, “Recent results on flocks, maximal exterior sets and inversive planes,” Combinatorics '88, 1, A. Barlotti et al. (eds.), Research and Lecture Notes in Mathematics, Mediterranean Press, 1991, 95-108.
2. L. Bader and G. Lunardon, “On the flocks of Q+(3, q),” Geom. Dedicata 29 (1989), 177-183. P1: rba Journal of Algebraic Combinatorics KL365-02-Johnson November 7, 1996 11:3 FLOCKS OF INFINITE HYPERBOLIC QUADRICS 51
3. R.D. Baker and G.L. Ebert, “A nonlinear flock in the Minkowski plane of order 11,” Congressus Num. 58 (1987), 75-81.
4. M. Biliotti and N.L. Johnson, “Bilinear flocks of quadratic cones,” J. Geom., to appear.
5. M. Biliotti and N.L. Johnson, “Variations on a theme of Dembowski,” Proceedings AMS Conference in Honor of J.G. Ostrom, Marcel Dekker, 1996 (to appear).
6. A. Bonisoli, “On the sharply 1-transitive subsets of PGL(2, pm ),” J. Geom. 31 (1988), 32-41.
7. R.P. Burn, “Bol quasifields and Pappus,” Theorem Math. Z. 105 (1968), 351-364.
8. F. De Clerck and H. Van Maldeghem, “Flocks of an infinite quadratic cone,” Bull. Belgian Math. Soc., Simon Stevin, 2 (1994), 399-415.
9. P. Dembowski, Finite Geometries, Berlin, Heidelberg, New York: Springer-Verlag, 1968.
10. J. Hanson and M.J. Kallaher, “Finite Bol quasifields are nearfields,” Utilitas Math. 37 (1990), 45-64.
11. V. Jha and N.L. Johnson, “Infinite flocks of quadratic cones,” J. Geom., to appear.
12. N.L. Johnson, “Flocks of hyperbolic quadrics and translation planes admitting affine homologies,” J. Geom. 34 (1989), 50-73.
13. M.J. Kallaher, “A note on Bol projective planes,” Arch. d. Math. 20 (1969), 329-333.
14. M.J. Kallaher, “A note on finite Bol quasifields,” Arch. d. Math. 23 (1972), 164-166.
15. W.F. Orr, “The Miquelian inversive plane I P(q) and the associated projective planes,” Ph.D. Thesis, Univ. Wisconsin, 1973.
16. R. Riesinger, “Faserungen, die aus Reguli mit einem gemeinsamen Geradenpaar zusammengesetzt sind,” J. Geom. 45 (1992), 137-157.
17. J.A. Thas, “Flocks of egglike inversive planes,” in Finite Geometric Structures and their Applications, A. Barlotti (Ed.), Rome, 1973, 189-191.
18. J.A. Thas, “Flocks of nonsingular ruled quadrics in PG(3, q),” Rend. Accad. Nat. Lincei 59 (1975), 83-85.
19. J.A. Thas, “Generalized quadrangles and flocks of cones,” Europ. J. Comb. 8 (1987), 441-452.
20. J.A. Thas, “Flocks, maximal exterior sets and inversive planes,” Contemp. Math. 111 (1990), 187-218.
21. J.A. Thas, “Recent results on flocks, maximal exterior sets and inversive planes,” Combinatorics '88, 1, A. Barlotti et al. (eds.), Research and Lecture Notes in Mathematics, Mediterranean Press, 1991, 95-108.