Symmetries of BLT-Sets

We do the tentative beginnings of a study of BLT-sets of generalised quadrangles via their symmetries. In particular, the study of whorls about a line leads us to hyperbolic reflections preserving a BLT-set of Q(4, q).     

Sets of Type (a, b) From Subgroups of ΓL(1, pR)

In this paper k-sets of type (a, b) with respect to hyperplanes are constructed in finite projective spaces using powers of Singer cycles. These are then used to construct further examples of sets of type (a, b) using various disjoint sets. The parameters of the associated strongly regular graphs are also calculated. The construction technique is then related to work of Foulser and Kallaher classifying rank three subgroups of AL(1, p ^R). It is shown that the sets of type (a, b) arising from the Foulser and Kallaher construction in the case of projective spaces are isomorphic to some of those constructed in the present paper.     

Relative symplectic subquadrangle hemisystems of the Hermitian surface

We introduce the notion of relative subquadrangle regular system of a generalized quadrangle. A relative subquadrangle regular system of order m on a generalized quadrangle S of order (s, t) is a set \({\mathcal R}\) of embedded subquadrangles with a prescribed intersection property with respect to a given subquadrangle T such that every point of S T lies on exactly m subquadrangles of \({\mathcal R}\) . If m is one half of the total number of such subquadrangles on a point we call \({\mathcal R}\) a relative subquadrangle hemisystem with respect to T. We construct two infinite families of symplectic relative subquadrangle hemisystems of the Hermitian surface \({{\mathcal H}(3,q^2)}\) , q even.     

A characterisation of thas maximal arcs in translation planes of square order

In 1974 J.A. Thas constructed a class of maximal arcs in certain translation planes of order q². We characterise these as being exactly those (non-trivial) maximal arcs that are stabilised by an homology of order q– 1.The first author gratefully acknowledges the support of an Australian Postgraduate Research Award.     

Flocks and ovals

An infinite family of q-clans, called the Subiaco q-clans, is constructed for q=2^e. Associated with these q-clans are flocks of quadratic cones, elation generalized quadrangles of order (q ², q), ovals of PG(2, q) and translation planes of order q ² with kernel GF(q). It is also shown that a q-clan, for q=2^e, is equivalent to a certain configuration of q+1 ovals of PG(2, q), called a herd.W. Cherowitzo gratefully acknowledges the support of the Australian Research Council and has the deepest gratitude and warmest regards for the Combinatorial Computing Research Group at the University of Western Australia for their congenial hospitality and moral support. I. Pinneri gratefully acknowledges the support of a University of Western Australia Research Scholarship.     

Hemisystems on the Hermitian Surface

The natural geometric setting of quadrics commuting with a Hermitiansurface of PG(3,q²), q odd, is adopted and a hemisystem on theHermitian surface H(3,q²) admitting the group P^–(4,q)is constructed, yielding a partial quadrangle PQ((q–1)/2,q²,(q–1)²/2) and a strongly regular graph srg((q³+1)(q+1)/2,(q²+1)(q–1)/2,(q–3)/2,(q–1)²/2).For q>3, no partial quadrangle or strongly regular graphwith these parameters was previously known, whereas when q=3,this is the Gewirtz graph. Thas conjectured that there are nohemisystems on H(3,q²) for q>3, so these are counterexamplesto his conjecture. Furthermore, a hemisystem on H(3,25) admitting3.A₇.2 is constructed. Finally, special sets (after Shult) andovoids on H(3,q²) are investigated.     

Configurations of ovals

We survey the known hyperovals in . We then survey the relationship of the study of configurations of ovals in called augmented fans to that of ovoids of , flocks of the quadratic cone of , flocks of a translation oval cone of and spreads of certain generalised quadrangles of order. We also consider the configurations of ovals in called herds and their relationship with flocks of the quadratic cone of.     

Tight sets and m-ovoids of finite polar spaces

An intriguing set of points of a generalised quadrangle was introduced in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] as a unification of the pre-existing notions of tight set and m-ovoid. It was shown in [J. Bamberg, M. Law, T. Penttila, Tight sets and m-ovoids of generalised quadrangles, Combinatorica, in press] that every intriguing set of points in a finite generalised quadrangle is a tight set or an m-ovoid (for some m). Moreover, it was shown that an m-ovoid and an i-tight set of a common generalised quadrangle intersect in mi points. These results yielded new proofs of old results, and in this paper, we study the natural analogue of intriguing sets in finite polar spaces of higher rank. In particular, we use the techniques developed in this paper to give an alternative proof of a result of Thas [J.A. Thas, Ovoids and spreads of finite classical polar spaces, Geom. Dedicata 10 (1-4) (1981) 135-143] that there are no ovoids of H(2r,q²), Q⁻(2r+1,q), and W(2r−1,q) for r>2. We also strengthen a result of Drudge on the non-existence of tight sets in W(2r−1,q), H(2r+1,q²), and Q⁺(2r+1,q), and we give a new proof of a result of De Winter, Luyckx, and Thas [S. De Winter, J.A. Thas, SPG-reguli satisfying the polar property and a new semipartial geometry, Des. Codes Cryptogr. 32 (1-3) (2004) 153-166; D. Luyckx, m-Systems of finite classical polar spaces, PhD thesis, The University of Ghent, 2002] that an m-system of W(4m+3,q) or Q⁻(4m+3,q) is a pseudo-ovoid of the ambient projective space.     

Hyperbolic Fibrations and q-Clans

We show there is a bijection between regular hyperbolic fibrations with constant back half and normalized q-clans. Thus there is also a bijection with flocks of a quadratic cone, once a conic of the flock has been specified. This yields a plethora of two-dimensional translation planes of even and odd order which arise from spreads admitting a regular elliptic cover.