Baer cone intersections

If K is a blocking set of a projective space and P a point not in K, then the projection of K from P onto a hyperplane of not containing P is a blocking set of . Projecting a blocking set K of PG(3, q²) from two different points P₀, P₁ onto a plane neither containing P₀ nor P₁, the intersection of the two cones with vertices P₀ and P₁ and bases the corresponding projections onto should constitute a big part of the blocking set. Looking for non-trivial blocking sets, the bases contain each a Baer subplane of . Hence the intersection of the two Baer cones over these Baer subplanes with the vertices P₀ and P₁ are part of the blocking set K in PG(3, q²). In this article, we describe the intersection configurations of two Baer cones in PG(3, q²).     

Blocking structures in finite projective planes

Exploring the classical Ceva configuration in a Desarguesian projective plane, we construct two families of minimal blocking sets as well as a new family of blocking semiovals in PG(2, 3^2h). Also, we show that these blocking sets of PG(2, q²), regarded as pointsets of the derived André plane , are still minimal blocking sets in . Furthermore, we prove that the new family of blocking semiovals in PG(2, 3^2h) gives rise to a family of blocking semiovals in the André plane as well.     

Dual blocking sets in projective and affine planes

A dual blocking set is a set of points which meets every blocking set but contains no line. We establish a lower bound for the cardinality of such a set, and characterize sets meeting the bound, in projective and affine planes.     

Projektive Einbettung nicht lokal projektiver Räume

It is well known that every locally projective linear space (M,M) with dimM 3, fulfilling the Bundle Theorem (B) can be embedded in a projective space. We give here a new construction for the projective embedding of linear spaces which need not be locally projective. Essentially for this new construction are the assumptions (A) and (C) that for any two bundles there are two points on every line which are incident with a line of each of these bundles. With the Embedding Theorem (7.4) of this note for example a [0,m]-space can be embedded in a projective space.

     

How many s-subspaces must miss a point set in PG(d, q)

The following result is well-known for finite projective spaces. The smallest cardinality of a set of points of PG(n, q) with the property that every s-subspace has a point in the set is (q ^n+1-s - 1)/(q - 1). We solve in finite projective spaces PG(n, q) the following problem. Given integers s and b with 0 ≤ s ≤ n - 1 and 1 ≤ b ≤ (q ^n+1-s - 1)/(q - 1), what is the smallest number of s-subspaces that must miss a set of b points. If d is the smallest integer such that b ≤ (q ^d+1 - 1)/(q - 1), then we shall see that the smallest number is obtained only when the b points generate a subspace of dimension d. We then also determine the smallest number of s-subspaces that must miss a set of b points of PG(n, q) which do not lie together in a subspace of dimension d. The results are obtained by geometrical and combinatorial arguments that rely on a strong algebraic result for projective planes by T. Szőnyi and Z. Weiner.     

Dominating Sets in Projective Planes

We describe small dominating sets of the incidence graphs of finite projective planes by establishing a stability result that shows that dominating sets are strongly related to blocking and covering sets. Our main result states that if a dominating set in a projective plane of order is smaller than (i.e., twice the size of a Baer subplane), then it contains either all but possibly one points of a line or all but possibly one lines through a point. Furthermore, we completely characterize dominating sets of size at most . In Desarguesian planes, we could rely on strong stability results on blocking sets to show that if a dominating set is sufficiently smaller than 3q, then it consists of the union of a blocking set and a covering set apart from a few points and lines.     

Line partitions of internal points to a conic in <Emphasis Type="Italic">PG</Emphasis>(2,<Emphasis Type="Italic">q</Emphasis>)

All sets of lines providing a partition of the set of internal points to a conic C in PG(2,q), q odd, are determined. There exist only three such linesets up to projectivities, namely the set of all non-tangent lines to C through an external point to C, the set of all non-tangent lines to C through a point in C, and, for square q, the set of all non-tangent lines to C belonging to a Baer subplane PG(2,√q) with √q+1 common points with C. This classification theorem is the analogous of a classical result by Segre and Korchmáros [9] characterizing the pencil of lines through an internal point to C as the unique set of lines, up to projectivities, which provides a partition of the set of all non-internal points to C. However, the proof is not analogous, since it does not rely on the famous Lemma of Tangents of Segre which was the main ingredient in [9]. The main tools in the present paper are certain partitions in conics of the set of all internal points to C, together with some recent combinatorial characterizations of blocking sets of non-secant lines, see [2], and of blocking sets of external lines, see [1].     

Blocking sets of Rédei type in projective Hjelmslev planes

In this paper, we introduce Rédei type blocking sets in projective Hjelmslev planes over finite chain rings. We construct, in Hjelmslev planes over chain rings of nilpotency index 2 that contain the residue field as a proper subring, the Baer subplanes associated with this subring as Rédei type blocking sets. Two further examples of Rédei type blocking sets are given for planes over Galois rings generalizing familiar constructions in projective planes over finite fields.     

4-Dimensional compact projective planes with a 5-dimensional nilradical

The article is a contribution to the classification of all 4-dimensional flexible compact projective planes. We assume that the collineation group is a 6-dimensional solvable Lie group which fixes some flag. If, moreover, the nilradical of the collineation group is 5-dimensional, then we get 4 families of new planes which are neither translation planes nor shift planes.Meinem Lehrer H. Salzmann zum 65. Geburtstag am 3.11.1995 in Dankbarkeit gewidmet     

Actions of almost simple groups on compact eight-dimensional projective planes are classical,almost

We show that the existence of an almost simple group of automorphisms of dimension greater than 10 characterizes the Hughes planes (including the quarternion plane) among the 8-dimensional compact projective planes.Dedicated to Prof. Helmut R. Salzmann on his 65th birthday     

Self-dual sets of type (m, n) in projective spaces

In this paper we introduce and analyze the notion of self-dual k-sets of type (m, n). We show that in a non-square order projective space such sets exist only if the dimension is odd. We prove that, in a projective space of odd dimension and order q, self-dual k-sets of type (m, n), with , are of elliptic and hyperbolic type, respectively. As a corollary we obtain a new characterization of the non-singular elliptic and hyperbolic quadrics.     

A characterization of the classical unital

We define Buekenhout unitals in derivable translation planes of dimension 2 over their kernel and provide a characterization of these unitals. We use this result to improve the characterization of classical unitals given by Lefèvre-Percsy [13] and Faina and Korchmáros [7].     

The translation planes of order 81 that admit SL(2,5), generated by affine elations

The translation planes of order 81 admitting SL(2, 5), generated by affine elations, are completely determined. There are seven mutually non-isomorphic translation planes, of which five are new. Each of these planes may be derived producing another set of seven mutually non-isomorphic translation planes admitting SL(2, 5), where the 3-elements are Baer. Of this latter set, five planes are new.     

Twisted derivations and distinct sets of lines that cover the same pairs of points

A partial projective plane of ordern consists of lines andn ² +n + 1 points such that every line hasn+1 points and distinct lines meet in a unique point. Suppose that two essentially different partial projective planes and of ordern, n a perfect square, that are defined on the same set of points cover the same pairs of points. For sufficiently largen we show that this implies that and have at leastn(n+1) lines. This bound is sharp and there exist essentially two different types of examples meeting the bound.As an application, we can show that derived planes provide an example for a pair of projective planes of square order with as much structure as possible in common, that is, as many lines as possible in common. Furthermore, we present a new method (twisted derivations) to obtain planes from one another by replacing the same number of lines as in a derivation.     

Note on the existence of large minimal blocking sets in galois planes

A subsetS of a finite projective plane of orderq is called a blocking set ifS meets every line but contains no line. For the size of an inclusion-minimal blocking setq+ +Sq +1 holds ([6]). Ifq is a square, then inPG(2,q) there are minimal blocking sets with cardinalityq +1. Ifq is not a square, then the various constructions known to the author yield minimal blocking sets with less than 3q points. In the present note we show that inPG(2,q),q1 (mod 4) there are minimal blocking sets having more thanqlog₂ q/2 points. The blocking sets constructed in this note contain the union ofk conics, whereklog₂ q/2. A slight modification of the construction works forq3 (mod 4) and gives the existence of minimal blocking sets of sizecqlog₂ q for some constantc.As a by-product we construct minimal blocking sets of cardinalityq +1, i.e. unitals, in Galois planes of square order. Since these unitals can be obtained as the union of parabolas, they are not classical.     

Unitals in the regular nearfield planes

Classes of parabolic unitals in the regular nearfield planes of odd square order are enumerated and classified. These unitals correspond to certain Buekenhout-Metz unitals in the classical plane. Their collineation groups are determined and the unitals are sorted by projective equivalence.      

Laguerre planes of even order and translation ovals

Translation Laguerre planes of even order are represented in high dimensional projective space over GF(2) by a collection of subspaces that satisfies a very simple condition.This research was supported for the respective authors by a grant from the University of Canterbury and by a Feodor Lynen Fellowship.     

Conics in finite projective planes

There does not exist a general theory of conics in finite projective planes, because the many definitions of conics which are equivalent in desarguesian projective planes yield different types of conics in more general situations. Thus even the use of the word conic can lead to confusion, particularly in the finite case. This note is an attempt to clarify these various definitions and give as an example in a finite projective plane a von Staudt conic which is not an Ostrom conic. We conjecture that any finite projective plane admitting an Ostrom conic must be desarguesian.     

Maximal partial spreads in PG (3, q)

We transfer the whole geometry of PG(3, q) over a non-singular quadric Q_4,q of PG(4, q) mapping suitably PG(3, q) over Q_4,q. More precisely the points of PG(3, q) are the lines of Q_4,q; the lines of PG(3, q) are the tangent cones of Q_4,q and the reguli of the hyperbolic quadrics hyperplane section of Q_4,q. A plane of PG(3, q) is the set of lines of Q_4,q meeting a fixed line of Q_4,q. We remark that this representation is valid also for a projective space over any field K and we apply the above representation to construct maximal partial spreads in PG(3, q). For q even we get new cardinalities for For q odd the cardinalities are partially known.