Kenneth L. Wantz

Hyperbolic Fibrations ofPG(3,q)

A hyperbolic fibration is set ofq?1 hyperbolic quadrics and two lines which together partition the points ofPG(3,q). The classical example of a hyperbolic fibration comes from a pencil of quadrics; however, several other families are known. In this paper we construct a new family of hyperbolic fibrations for odd prime powersq.As an application of hyperbolic fibrations, we note that they can be used to construct 2q?1(not necessarily inequivalent) spreads ofPG(3,q) by choosing one ruling family from each of the hyperbolic quadrics in the fibration. For our new fibration we discuss some properties of the spreads obtained in the above manner.

Baer subgeometry partitions

For any odd integern ≥3 and prime powerq, it is known thatPG(n−1, q2) can be partitioned into pairwise disjoint subgeometries isomorphic toPG(n−1, q) by taking point orbits under an appropriate subgroup of a Singer cycle ofPG(n−1, q2). In this paper, we construct Baer subgeometry partitions of these spaces which do not arise in the classical manner. We further illustrate some of the connections between Baer subgeometry partitions and several other areas of combinatorial interest, most notably projective sets and flagtransitive translation planes.

Generalized Pellegrino caps

Abstract A cap in a projective or affine geometry is a set of points with the property that no line meets the set in more than two points. Barwick et al. [S.G. Barwick, W.-A. Jackson, C.T. Quinn, Conics and caps, J. Geom. 100 (2011) 15–28] provide a construction of caps in PG ( 4 , q ) by “lifting” arbitrary caps of PG ( 2 , q 2 ) , such as conics. In this article, we extend this construction by considering when the union of two or more conics in AG ( 2 , q 2 ) can be lifted to a cap of AG ( 4 , q ) using a similar coordinate transformation. In particular, the authors investigate a family of caps of size 2 ( q 2 + 1 ) in AG ( 4 , q ) for all prime powers q > 2 , of which the celebrated Pellegrino 20-cap in AG ( 4 , 3 ) is the smallest example.

A New Class of Unitals in the Hughes Plane

A new class of unitals in the Hughes planes is enumerated and classified. The unital obtained by L. A. Rosati is shown to be a member of this class. Their collineation groups are determined and the unitals are sorted by projective equivalence. The dual designs are described and certain members are shown to be self-dual.

Enumeration of orthogonal Buekenhout unitals

In this paper we develop general techniques for enumerating orthogonal Buekenhout unitals embedded in two-dimensional translation planes. We then apply these techniques in the regular nearfield planes, the odd-order Hall planes, and the odd-order flag-transitive affine planes. Stabilizers of the resulting unitals also are computed.