Applications of Number Theory to Ovoids and Translation Planes

In this paper we show that for each prime p7 there exists a translation plane of order p ² of Mason–Ostrom type. These planes occur as six-dimensional ovoids being projections of the eight-dimensional binary ovoids of Conway, Kleidman and Wilson. In order to verify the existence of such projections we prove certain properties of two particular quadratic forms using classical methods form number theory.     

A characterization of quaternion planes

The eight-dimensional planes admitting SL₂ as a group of automorphisms are determined.Dedicated to my teacher, Prof. H. Salzmann, on his 60th birthday     

Numerical Burniat and Irregular Surfaces

In this paper, we construct three numerical Burniat surfaces as desingularizations of double planes of degree >10. Two are surfaces having the bigenus P ₂=4 and the third is a surface having the bigenus P ₂=5. In addition, another surface of general type is constructed as a desingularization of a double plane of degree 12 having the birational invariants: q=p _g =1, P ₂=4. One of the numerical Burniat surfaces with P ₂=4 is obtained as a desingularization of a double plane of degree 22 with an irreducible branch locus, so it is a good candidate for having torsion zero. Moreover, its bicanonical transformation seems to be birational.     

Lower bounds for coverings of pairs by large blocks

Letnkt be positive integers, andX—a set ofn elements. LetC(n, k, t) be the smallest integerm such that there existm k-tuples ofX B ₁ B ₂,...,B _m with the property that everyt-tuple ofX is contained in at least oneB _i . It is shown that in many cases the standard lower bound forC(n, k, 2) can be improved (k sufficiently large,n/k being fixed). Some exact values ofC(n, k, 2) are also obtained.     

Cyclic Ostrom Spreads

We call a transitive permutation group G stable if every non-trivial orbit B, of a point stabilizerG _O , is such that some element of G-G _O leaves invariant B {O}. We characterize all finite affine planes of order n that admit a collineation group G that acts, on the points of , as a stable 3/2-transitive permutation group of rank n+2 . The planes obtained are precisely what we have called cyclic Ostrom planes; these are translation planes whose associated spreads are obtained from the Desarguesian spread , of order n=p^r , by replacing a partial subspread of by another consisting of GF(p) subspaces that are unions of kern orbits. All André spreads, as well as many other spreads, are examples of cyclic Ostrom spreads.     

Large Almost Simple Groups Acting on 16-Dimensional Compact Projective Planes

 This paper deals with the so-called Salzmann program aiming to classify special geometries according to their automorphism groups. Here, topological connected compact projective planes are considered. If finite-dimensional, such planes are of dimension 2, 4, 8, or 16. The classical example of a 16-dimensional, compact projective plane is the projective plane over the octonions with 78-dimensional automorphism group E₆(−26). A 16-dimensional, compact projective plane ? admitting an automorphism group of dimension 41 or more is clasical, [23] 87.5 and 87.7. For the special case of a semisimple group Δ acting on ? the same result can be obtained if dim , see [22]. Our aim is to lower this bound. We show: if Δ is semisimple and dim , then ? is either classical or a Moufang-Hughes plane or Δ is isomorphic to Spin₉ (ℝ, r), r∈{0, 1}. The proof consists of two parts. In [16] it has been shown that Δ is in fact almost simple or isomorphic to SL₃?ċSpin₃ℝ. In the underlying paper we can therefore restrict our considerations to the case that Δ is almost simple, and the corresponding planes are classified. Received 10 February 1997; in final form 19 December 1997     

A note on Hilbert and Beltrami systems

Hilbert and Beltrami (line- ) systems were introduced by H. Mohrmann, Math. Ann. 85 (1922) p.177- 183. These systems give examples of non- desarguesian affine planes, in fact, the earliest known examples are of this type. We describe a construction for “generalized Beltrami systems”, and show that every such system defines a topological affine plane with point set ?². Since our construction uses only the topological structure of ?²- planes, it is possible to iterate this process. As an application, we obtain an embeddability theorem for a class of two- dimensional stable planes, including Strambach’s exceptional SL₂R- plane.     

The skew-hyperbolic motion group of the quaternion plane

Up to conjugation, there exist three different polarities of the projective plane over Hamilton's quaternions . The skew hyperbolic motion group of P₂ is introduced as the centralizer of a polarity of the third kind. According to a result of R. Löwen, the quaternion plane is characterized among the eight-dimensional stable planes by the fact that it admits an effective action of the centralizer of a polarity of the first or second kind (i.e., the elliptic or the hyperbolic motion group). In the present paper, we prove the analogous result for skew hyperbolic case.     

On Planes of Order p2 in Which Every Quadrangle Generates a Subplane of Order p

After Gleason's result, in the late fifties the following conjecture appeared: if in a finite projective plane every quadrangle is contained in a unique Desarguesian proper subplane of order p, then the plane is Desarguesian (and its order is p ^d for some d). In this paper we prove the conjecture in the case when the plane is of order p ² and p is a prime.     

Least Area Planes in Gromov Hyperbolic 3-Spaces with Co-compact Metric

Let M be a closed irreducible Riemannian 3-manifold such that π₁(M) is word hyperbolic, and p: X→ M the universal covering. Suppose that X has the Riemannian metric induced form that on M via p. In this paper, we will show that any Jordan curve in the boundary ∂X of X spans a properly embedded least area plane in X.Mathematics Subject Classiffications (2000). Primary 57M50; secondary 53A10     

Realizations of Regular Toroidal Maps of Type {4,4}

We determine and completely describe all pure realizations of the finite toroidal maps of types {4,4} _(b,0) and {4,4} _(b,b) , b \geq; 2 . For large values of b , most such realizations are eight-dimensional. Received January 7, 1999, and in revised form May 17, 1999. Online publication May 16, 2000.     

Inherited conics in Hall planes

The existence of ovals and hyperovals is an old question in the theory of non-Desarguesian planes. The aim of this paper is to describe when a conic of $\mathrm{PG}(2,q)$ remains an arc in the Hall plane obtained by derivation. Some combinatorial properties of the inherited conics are obtained also in those cases when it is not an arc. The key ingredient of the proof is an old lemma by Segre–Korchmáros on Desargues configurations with perspective triangles inscribed in a conic.     

Classification of Embeddings of the Flag Geometries of Projective Planes in Finite Projective Spaces, Part 2

The flag geometry Γ=( ,  , I) of a finite projective plane Π of order s is the generalized hexagon of order (s, 1) obtained from Π by putting equal to the set of all flags of Π, by putting equal to the set of all points and lines of Π, and where I is the natural incidence relation (inverse containment), i.e., Γ is the dual of the double of Π in the sense of H. Van Maldeghem (1998, “Generalized Polygons,” Birkhäuser Verlag, Basel). Then we say that Γ is fully and weakly embedded in the finite projective space PG(d, q) if Γ is a subgeometry of the natural point-line geometry associated with PG(d, q), if s=q, if the set of points of Γ generates PG(d, q), and if the set of points of Γ not opposite any given point of Γ does not generate PG(d, q). In two earlier papers we have shown that the dimension d of the projective space belongs to {6, 7, 8}, that the projective plane Π is Desarguesian, and we have classified the full and weak embeddings of Γ (Γ as above) in the case that there are two opposite lines L, M of Γ with the property that the subspace U_L, M of PG(d, q) generated by all lines of Γ meeting either L or M has dimension 6 (which is automatically satisfied if d=6). In the present paper, we partly handle the case d=7; more precisely, we consider for d=7 the case where for all pairs (L, M) of opposite lines of Γ, the subspace U_L, M has dimension 7 and where there exist four lines concurrent with L contained in a 4-dimensional subspace of PG(7, q).     

Quasi-affine symmetric designs

A symmetric design with parameters v = q ²(q + 2), k = q(q + 1), λ = q, q ≥ 2, is called a quasi-affine design if its point set can be partitioned into q + 2 subsets P ₀, P ₁,..., P _q , P _q+1 such that the induced structure in every point neighborhood is an affine plane of order q (repeated q times). A quasi-affine design with q ≥ 3 determines its point neighborhoods uniquely and dual of such a design is also a quasi-affine design. These structural properties pave way for definition of a strongly quasi-affine design and it is also shown that associated with every quasi-affine design is a unique strongly quasi-affine design from which the given quasi-affine design is obtained by certain unique cutting and pasting operation. This investigation also enables us to associate a unique 2-regular graph with q + 2 vertices and in turn, a unique colored partition of the integer q + 2. These combinatorial consequences are finally used to obtain an exponential lower bound on the number of non-isomorphic solutions of such symmetric designs improving the earlier lower bound of 2. Work of Sanjeevani Gharge is supported by Faculty Improvement Programme of U.G.C., India.     

On blocking sets of inversive planes

Let S be a blocking set in an inversive plane of order q. It was shown by Bruen and Rothschild 1 that |S| ≥ 2q for q ≥ 9. We prove that if q is sufficiently large, C is a fixed natural number and |S = 2q + C, then roughly 2/3 of the circles of the plane meet S in one point and 1/3 of the circles of the plane meet S in four points. The complete classification of minimal blocking sets in inversive planes of order q ≤ 5 and the sizes of some examples of minimal blocking sets in planes of order q ≤ 37 are given. Geometric properties of some of these blocking sets are also studied. © 2004 Wiley Periodicals, Inc.     

Unitals in Finite Desarguesian Planes

We show that a suitable 2-dimensional linear system of Hermitian curves of PG(2,q ²) defines a model for the Desarguesian plane PG(2,q). Using this model we give the following group-theoretic characterization of the classical unitals. A unital in PG(2,q ²) is classical if and only if it is fixed by a linear collineation group of order 6(q + 1)² that fixes no point or line in PG(2,q ²).     

A non-miquelian Laguerre plane satisfying a theorem of miquelian type

This note shows that a theorem of miquelian type known as (M2) holds in a certain non miquelian Laguerre plane of shear type as defined by Löwen and Pfüller[1].Dedicated to Professor H. Karzel on the occasion of his 70th birthday     

Seven Cubes and Ten 24-Cells

The 45 diagonal triangles of the six-dimensional polytope 2 ₂₁ (representing the 45 tritangent planes of the cubic surface) are the vertex figures of 45 cubes { 4,3} inscribed in the seven-dimensional polytope 3 ₂₁ , which has 56 vertices. Since 45 x 56 = 8 x 315 , there are altogether 315 such cubes. They are the vertex figures of 315 specimens of the four-dimensional polytope { 3,4,3 } , which has 24 vertices. Since 315 x 240 = 24 x 3150 , there are altogether 3150 { 3,4,3 } 's inscribed in the eight-dimensional polytope 4 ₂₁ . They are the vertex figures of 3150 four-dimensional honeycombs { 3,3,4,3 } inscribed in the eight-dimensional honeycomb 5 ₂₁ . In other words, each point of the ₈ lattice belongs to 3150 inscribed ₄ lattices of minimal size. Analogously, in unitary 4 -space there are 3150 regular complex polygons 3 { 4 } 3 inscribed in the Witting polytope 3 { 3 } 3 { 3 } 3 { 3 } 3 . Received March 12, 1996, and in revised form May 17, 1996.     

Stabilizers of collineation groups of smooth stable planes

Let be a smooth stable plane of dimension n (see Definition 1.2) and let Δ be a closed subgroup of the collineation group of which fixes some point p. We derive some results on the group-theoretical structure of Δ, e.g. that Δ is a linear Lie group (Theorem 3.7). As a by-product this shows that no (affine or projective) Moulton plane can be turned into a smooth plane. If Δ fixes some flag, then any Levi subgroup Ψ of Δ is a compact group and Δ is contained in the flag stabilizer of the classical Moufang plane of dimension n (Corollary 3.1 and Theorem 3.7). Let Δ fix three concurrent lines through the point p. If is one of the classical projective planes over the reals, the complex numbers, the quaternions, or the Cayley numbers, then the dimension of Δ is d_class = 3, 6, 15, or 38, respectively. We show that for a smooth stable (projective) plane S of dimension 2l either S is an almost projective translation plane (classical projective plane) or that dim Δ ≤ d_class − l holds (Theorems 4.1 and 4.2).