Sublines of Prime Order Contained in the Set of Internal Points of a Conic

AbstractIn [2] it was shown that if q ≥ 4n2−8n+2 then there are no subplanes of order q contained in the set of internal points of a conic in PG(2,qn), q odd, n≥ 3. In this article we improve this bound in the case where q is prime to $q > 2n^2-(4-2\sqrt{3})n+(3-2\sqrt{3})$, and prove a stronger theorem by considering sublines instead of subplanes. We also explain how one can apply this result to flocks of a quadratic cone in PG(3,qn), ovoids of Q(4,qn), rank two commutative semifields, and eggs in PG(4n−1,q).

[1]  Tim Penttila,et al.  On Eggs and Translation Generalised Quadrangles , 2001, J. Comb. Theory, Ser. A.

[2]  J. W. P. Hirschfeld.  FINITE GENERALIZED QUADRANGLES (Research Notes in Mathematics, 110) , 1985 .

[3]  Joseph A. Thas,et al.  Generalized Quadrangles and Flocks of Cones , 1987, Eur. J. Comb..

[4]  Michel Lavrauw,et al.  On the classification of semifield flocks , 2003 .

[5]  Simeon Ball,et al.  Commutative semifields of rank 2 over their middle nucleus , 2002 .

[6]  Leonard Eugene Dickson.  On commutative linear algebras in which division is always uniquely possible , 1906 .

[7]  Joseph A. Thas,et al.  Translation Ovoids of Generalized Quadrangles and Hexagnos , 1998 .

[8]  Guglielmo Lunardon,et al.  A New Semifield Flock , 1999, J. Comb. Theory, Ser. A.

[9]  Tim Penttila,et al.  Ovoids of Parabolic Spaces , 2000 .

[10]  Michel Lavrauw,et al.  Scattered spaces with respect to spreads, and eggs in finite projective spaces : Scattered subspaces with respect to spreads, and eggs in finite projective spaces , 2001 .

[11]  Joseph A. Thas.  Generalized Quadrangles of Order (s, s2), II , 1997, J. Comb. Theory, Ser. A.

[12]  Simeon Ball,et al.  The six semifield planes associated with a semifield flock , 2004 .

[13]  P. Dembowski.  Finite geometries , 1997 .

[14]  H. Gevaert,et al.  Flocks of quadratic cones, generalized quadrangles and translation planes , 1988 .

[15]  J. Hirschfeld.  Projective Geometries Over Finite Fields , 1980 .

[16]  Stephen D. Cohen,et al.  Commutative semifields, two dimensional over their middle nuclei , 1982 .

[17]  G. Lunardon,et al.  Flocks, Ovoids of Q(4,q)and Designs , 1997 .

[18]  Stanley E. Payne.  An essay on skew translation generalized quadrangles , 1989 .