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Ovoids of PG(3,q) stabilized by a central collineation

Matthew R Brown
2003 European journal of combinatorics (Print)  
In this paper we give a short proof that an ovoid of PG(3, q) is stabilized by a non-trivial central collineation of PG(3, q) if and only if it is an elliptic quadric.  ...  Acknowledgement The author acknowledges the support of the University of Ghent grant GOA 12050300.  ...  An ovoid of PG(3, q) is stabilized by a central collineation if and only if it is an elliptic quadric.  ... 
doi:10.1016/s0195-6698(03)00028-3 fatcat:mksray4gjrerxlfpy3f334mola

Regular Packings of PG (3,q)

T. Penttila, B. William
1998 European journal of combinatorics (Print)  
Two regular packings of PG(3, q) are constructed whenever q ≡ 2 (mod 3), with each packing admitting a cyclic group of order q 2 +q + 1 acting regularly on the regular spreads in the packing.  ...  The resulting families of translation planes of order q 4 include the Lorimer-Rahilly and Johnson-Walker planes of order 16.  ...  Now G acts faithfully on PG(3, q)/P or on π (as G S contains no nontrivial central collineations), so G ∼ =Ḡ ≤ PG L(3, q): letr be the resulting image in PG L(3, q) of r .  ... 
doi:10.1006/eujc.1998.0239 fatcat:nxeyvnlsm5c3nonjhmtmm75gu4

Group theoretic characterizations of Buekenhout–Metz unitals in $\mathop{\mathrm{PG}}(2,q^{2})$

Giorgio Donati, Nicola Durante
2010 Journal of Algebraic Combinatorics  
Let G be the group of projectivities stabilizing a unital U in PG(2, q 2 ) and let A, B be two distinct points of U.  ...  In this paper we prove that, if G has an elation group of order q with center A and a group of projectivities stabilizing both A and B of order a divisor of q − 1 greater than 2( √ q − 1), then U is an  ...  Next we recall that a central collineation of PG(2, q 2 ) is a collineation α fixing every point of a line (the axis of α) and fixing every line through a point C (the center of α).  ... 
doi:10.1007/s10801-010-0250-8 fatcat:ghlhjk4ma5btzeovcsw6zftd7i

On the nonexistence of certain Hughes generalized quadrangles

Joris De Kaey, Alan Offer, Hendrik Van Maldeghem
2007 Designs, Codes and Cryptography  
that every central elation of the subquadrangle is induced by a collineation of Γ.  ...  In this paper, we prove that the Hermitian quadrangle H(4, q 2 ) is the unique generalized quadrangle Γ of order (q 2 , q 3 ) containing some subquadrangle of order (q 2 , q) isomorphic to H(3, q 2 ) such  ...  In particular, if g is a central collineation in ∆, then it is also a central collineation in Γ. Proof. The stabilizer G O acts on O as PGU 3 (q).  ... 
doi:10.1007/s10623-007-9066-5 fatcat:mpg2crfalnb7pmysafkvfs7nrm

The (56, 11, 2) design of Hall, Lane, and Wales

W Jónsson
1973 Journal of combinatorial theory. Series A  
The hint for this construction comes from the remark in [3] that the points of D can be looked upon as a set of ovoids in the projective plane PG(2,4) upon which PSL(3, 4) acts.  ...  Using the representation of D derived from S" it is possible to compute that D has a group of collineations which is isomorphic to a subgroup of index 3 in the group of all collineations and correlations  ...  (3, 4) extended by an element of order 2 which in a suitable coordinate system of PG (2, 4) corresponds to a field automorfihism of GF(4).  ... 
doi:10.1016/0097-3165(73)90068-x fatcat:3pst3d3mhjh73esg4cqhvv3shu

Page 2223 of Mathematical Reviews Vol. , Issue 2004c [page]

2004 Mathematical Reviews  
De Clerck (B-GHNT-PM; Ghent) 2004c:51006 51E20 05B25 Brown, Matthew R. (5-ADLD-SPM; Adelaide) Ovoids of PG(3, ¢) stabilized by a central collineation. (English summary) European J.  ...  An ovoid of PG(3,q) is a set of g? +1 points no three of which are collinear. An elliptic quadric is such an object, and the converse is true for g odd, and for g = 4,16.  ... 

Codes from Generalized Hexagons

A. De Wispelaere, H. Van Maldeghem
2005 Designs, Codes and Cryptography  
These in turn are constructed using a new distance-2-ovoid of the classical generalized hexagon H(4).  ...  In fact, the two-character set is the union of two orbits in PG(5, 4) under the action of L 2 (13).  ...  Table 3 : Dual codes related to the generalized hexagons of order 2. Looking at the tables above, one might try to generalize most of these results to arbitrary H(q) and/or its dual.  ... 
doi:10.1007/s10623-005-4035-3 fatcat:dzvjcui3kjen7prowy4twaks7y

Uniform Hyperplanes of Finite Dual Polar Spaces of Rank 3

A. Pasini, S. Shpectorov
2001 Journal of combinatorial theory. Series A  
We say that a hyperplane H of 2 is locally singular (respectively, quadrangular or ovoidal) if H & Q is the perp of a point (resp. a subquadrangle or an ovoid) of Q for every quad Q of 2.  ...  Let 2 be a finite thick dual polar space of rank 3.  ...  ACKNOWLEDGMENT The authors thank Ernest Shult for having pointed out a mistake in an earlier version of this paper and for having provided them with Proposition 2.8 and its proof.  ... 
doi:10.1006/jcta.2000.3136 fatcat:6gnscybpbvg7zbb7yzwjbchxbq

Classification of 8-dimensional rank two commutative semifields [article]

Michel Lavrauw, Morgan Rodgers
2016 arXiv   pre-print
This is done using computational methods utilizing the connection to linear sets in PG(2,q^4).  ...  The implications of these results are detailed for other geometric structures such as semifield flocks, ovoids of parabolic quadrics, and eggs.  ...  An ovoid O in Q(4, s) is a translation ovoid if there is a point p ∈ O and a group G of collineations of Q(4, s) stabilizing O, fixing p, and acting regularly on the points of O \{p}.  ... 
arXiv:1606.06151v2 fatcat:pdlefywj25cg5lxnnpc3metlbe

An unusual generalized quadrangle of order sixteen

Stanley E Payne, James E Conklin
1978 Journal of combinatorial theory. Series A  
The full collineation group of 3 is determined.  ...  The generalized quadrangle 9 associated by the method of Tits with the nonconical ovoid discovered by M. Hall, Jr., in pG(2, 16) is shown to contain no subquadrangle of order 4.  ...  In Section 3 the full collineation group of 9' is determined, with a by-product being that the group of collineations of PG (2, 16) leaving L&r' fixed setwise is exactly the subgroup found by Hall in  ... 
doi:10.1016/0097-3165(78)90044-4 fatcat:twr7g3vxjzdyzmhqkf2hlhistq

Finite elation Laguerre planes admitting a two-transitive group on their set of generators

Günter Steinke, Markus Stroppel
2013 Innovations in Incidence Geometry Algebraic Topological and Combinatorial  
Equivalently, each translation generalized quadrangle of order q with a group of automorphisms acting two-transitively on the set of lines through the base point is classical.  ...  We investigate finite elation Laguerre planes admitting a group of automorphisms that is two-transitive on the set of generators.  ...  The present investigation has been conducted during a stay of the second author as a Visiting Erskine Fellow at the University of Canterbury, Christchurch, New Zealand.  ... 
doi:10.2140/iig.2013.13.207 fatcat:vtzvbqjitrbcpbdmzljes7j22a

Page 4584 of Mathematical Reviews Vol. , Issue 94h [page]

1994 Mathematical Reviews  
De Clerck, Flocks and 51 GEOMETRY 4584 partial flocks of the quadratic cone in PG(3,q) (379-393); J. van Bon, Some extended generalized hexagons (395-403); F.  ...  For every vertical line g of N the mapping 6,:N — N defined by 6,(P) = (P,N 8): (PN g), is a collineation of N, which is called a Bol reflection.  ... 

On the existence of O'Nan configurations in Buekenhout unitals in PG(2,q^2) [article]

Tao Feng, Weicong Li
2018 arXiv   pre-print
In this paper, we establish the existence of O'Nan configurations in all nonclassical ovoidal Buekenhout-Metz unitals in PG(2,q^2).  ...  Acknowledgement This work was supported by National Natural Science Foundation of China under Grant No. 11771392. Reference  ...  The odd characteristic case Consider the following involutionary central collineation of PGL(3, q 2 ): σ : (x, y, z) → (−x, y, z).  ... 
arXiv:1810.00705v2 fatcat:ztdfjj76ane5rnxunl4i3wn4ea

Contents

2003 European journal of combinatorics (Print)  
-O., On the Eulerian polynomials of type D . . . . . . 391 BROWN, M.R., Ovoids of PG(3, q) stabilized by a central collineation . 409 MERRIS, R., Split graphs . . . . . . . . . . . . . 413 BOBEN, M. and  ...  -Q. and WANG, K., s-Regular cyclic coverings of the threedimensional hypercube Q 3 . . . . . . . . . . . . 719 DELANOTE, M., Point derivation of partial geometries . . . . . . 733 WILDBERGER, N.J., Minuscule  ... 
doi:10.1016/s0195-6698(03)00149-5 fatcat:4nmkuxvgbzgr7nicrcrobdwvoa

Index

2003 European journal of combinatorics (Print)  
tree-like equalities . . . . . . . 557 BROWN, M.R., Ovoids of PG(3, q) stabilized by a central collineation . 409 BROWN, M.R., DE BEULE, J. and STORME, L., Maximal partial spreads of T 2 (O) and  ...  -Q. and WANG, K., s-Regular cyclic coverings of the threedimensional hypercube Q 3 . . . . . . . . . . . 719 FIORINI, S., A combinatorial study of partial order polytopes . . . . 149 FRANCHI, C. and VSEMIRNOV  ... 
doi:10.1016/s0195-6698(03)00150-1 fatcat:q6zy3wos7vdkzfy55qdwge5nl4
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