On multiple blocking sets in Galois planes

This article continues the study of multiple blocking sets in PG(2,q). In [A. Blokhuis, L. Storme, T. Szőnyi, Lacunary polynomials, multiple blocking sets and Baer subplanes.J. London Math. Soc. (2)60(1999), 321–332. MR1724814 (2000j:05025) Zbl 0940.51007], using lacunary polynomials, it was proven thatt-fold blocking sets of PG(2,q),qsquare,t<q¼/2, of size smaller thant(q+ 1) +cqq⅔, withcq= 2−⅓whenqis a power of 2 or 3 andcq= 1 otherwise, contain the union oftpairwise disjoint Baer subplanes whent≥ 2, or a line or a Baer subplane whent= 1. We now combine the method of lacunary polynomials with the use of algebraic curves to improve the known characterization results on multiple blocking sets and to prove at(modp) result on smallt-fold blocking sets of PG(2,q=pn),pprime,n≥ 1.

On some hyperbolic planes from finite projective planes

LetΠ=(P,L,I)be a finite projective plane of ordern, and letΠ′=(P′,L′,I′)be a subplane ofΠwith ordermwhich is not a Baer subplane (i.e.,n≥m2+m). Consider the substructureΠ0=(P0,L0,I0)withP0=P\{X∈P|XIl, l∈L′},L0=L\L′whereI0stands for the restriction ofItoP0×L0. It is shown that everyΠ0is a hyperbolic plane, in the sense of Graves, ifn≥m2+m+1+m2+m+2. Also we give some combinatorial properties of the line classes inΠ0hyperbolic planes, and some relations between its points and lines.

Pseudo-Homogeneous Coordinates for Hughes Planes

Among the projective planes, the class of Hughes planes has received much interest, for several good reasons. However, the existing descriptions of these planes are somewhat unsatisfactory. We introduce pseudo-homogeneous coordinates which at the same time are easy to handle and give insight into the action of the group that is generated by all elations of the desarguesian Baer subplane of a Hughes plane. The information about the orbit decomposition is then used to give a description in terms of coset spaces of this group. Finally, we exhibit a non-closing Desargues configuration in terms of coordinates.

Divisible Semiplanes, Arcs, and Relative Difference Sets

In this paper we shall be concerned with arcs of divisible semiplanes. With one exception, all known divisible semiplanes D (also called “elliptic” semiplanes) arise by omitting the empty set or a Baer subset from a projective plane Π, i.e., D = Π\S, where S is one of the following:(i) S is the empty set.(ii) S consists of a line L with all its points and a point p with all the lines through it.(iii) S is a Baer subplane of Π.We will introduce a definition of “arc” in divisible semiplanes; in the examples just mentioned, arcs of D will be arcs of Π that interact in a prescribed manner with the Baer subset S omitted. The precise definition (to be given in Section 2) is chosen in such a way that divisible semiplanes admitting an abelian Singer group (i.e., a group acting regularly on both points and lines) and then a relative difference set D will always contain a large collection of arcs related to D (to be precise, —D and all its translates will be arcs).

Characterizations of the Generalized Hughes Planes

Let be a projective plane and a subplane of . If l is a line of , we let denote the group of all elations in that have as axis and leave Q invariant. In [12, p. 921], Ostrom asked for a description of all finite planes that have a Baer subplane with the property that for all lines l of . Here denotes the order of G. Both the desarguesian planes of square order and the generalized Hughes planes have this property (Hughes [10], Ostrom [14], Dembowski [6]). One of the aims of this paper is to show that these are the only planes having such a Baer subplane.

A note on generalized Hall planes

We prove that if π is a generalized Hall plane of odd order with associated Baer subplane π0 then π is a Hall plane if and only if there is a collineation σ of π such that π0σ ∩ π0 is an affine point.

A characterization of the Hall planes of odd order

The Hall projective planes of odd order are characterized in terms of their translations, collineations which fix all the points of a Baer subplane, and involutory homologies.