Pappian Projective Plane Research Articles

Embeddings of hermitian unitals into pappian projective planes

Every embedding of a hermitian unital with at least four points on a block into any pappian projective plane is standard, i.e. it originates from an inclusion of the pertinent fields. This result about embeddings also allows us to determine the full automorphism groups of (generalized) hermitian unitals.

The class of non-Desarguesian projective planes is Borel complete

For every infinite graph $\Gamma$ we construct a non-Desarguesian projective plane $P^*_{\Gamma}$ of the same size as $\Gamma$ such that $Aut(\Gamma) \cong Aut(P^*_{\Gamma})$ and $\Gamma_1 \cong \Gamma_2$ iff $P^*_{\Gamma_1} \cong P^*_{\Gamma_2}$. Furthermore, restricted to structures with domain $\omega$, the map $\Gamma \mapsto P^*_{\Gamma}$ is Borel. On one side, this shows that the class of countable non-Desarguesian projective planes is Borel complete, and thus not admitting a Ulm type system of invariants. On the other side, we rediscover the main result of [15] on the realizability of every group as the group of collineations of some projective plane. Finally, we use classical results of projective geometry to prove that the class of countable Pappian projective planes is Borel complete.

∏ 1 1 -Completeness of the Computable Categoricity Problem for Projective Planes

Computable presentations for projective planes are studied. We prove that the problem of computable categoricity is ∏ 1 1 -complete for the following classes of projective planes: Pappian projective planes, Desarguesian projective planes, arbitrary projective planes.

The Theory of Projective Planes is Complete with Respect to Degree Spectra and Effective Dimensions

We prove that the theory of Pappian projective planes is complete with respect to degree spectra of automorphically nontrivial structures, effective dimensions, degree spectra of relations, categoricity spectra, and automorphism spectra. Therefore, for every natural n ≥ 2, there exists a computable Pappian projective plane with computable dimension n.

Complexity of Isomorphism Problem for Computable Projective Planes

We study computable presentations for projective planes. We show that the isomorphism problem is Σ 1 1 complete for Pappian projective planes, Desarguesian projective planes, and arbitrary projective planes. Bibliography: 11 titles.

Noncomputability of classes of pappian and desarguesian projective planes

We study computable representations of projective planes and prove that the class of all pappian projective planes and the class of all desarguesian projective planes have no computable numberings (up to computable isomorphism).

Sharply [formula omitted]-transitive groups of projectivities in generalized polygons

The group of projectivities of (a line of) a projective plane is always 3-transitive. It is well known that the projective planes with a sharply 3-transitive group of projectivities are classified: they are precisely the Pappian projective planes. It is also well known that the group of projectivities of a generalized polygon is 2-transitive. Here, we classify all generalized quadrangles, all finite generalized hexagons, and the parameter sets of all finite generalized octagons with a sharply 2-transitive group of projectivities.

Epimorphisms of Generalized Polygons Part 1: GeometricalCharacterizations

A generalized polygon is a thick incidence geometry of rank 2 such that the girth of the incidence graph is twice the diameter of the incidence graph. These geometries are introduced by Tits [17] for group-theoretical purposes, but became an interesting research object in their own right. For an overview of the geometric study, see [19]. Obviously, the notion of morphism is an important one when dealing with geometries. For instance, monomorphisms are equivalent to embeddings of one geometry into the other (and on the group-theoretical level often give rise to maximal subgroups); isomorphisms clearly are needed to distinguish new geometries from old ones, but also to determine automorphism groups; epimorphisms can be used to construct quotient geometries or cover geometries (as the geometric counterpart of local fields). But in all these cases, the geometries considered in the literature are of the same kind, i.e., they have same gonality and diameter. In the present paper, we initiate the study of morphisms between generalized polygons of unequal gonality. We restrict ourselves to epimorphisms since we are motivated by some nice examples in this case. The study of monomorphisms and embeddings requires different techniques. To see the problem, a good starting point is Pasini’s theorem [13] that states that any epimorphism between two generalized n-gons, n > 2, is either an isomorphism or has infinite fibers. In particular, if an epimorphism is bijective if restricted to one point row, then it is a global isomorphism. This is no longer true for epimorphisms from a generalized m-gon to an n-gon, m 6= n, and a standard example is given is Section 3 below. It describes an epimorphism from the classical split Cayley hexagon over some field F to the ordinary Pappian projective plane over F with the property that line pencils and point rows are mapped bijectively onto line pencils and point rows, respectively. The paper is organized as follows. In section 2, we propose a quite general classification system for epimorphisms of geometries, from the local point of view. In the next section, Section 3, we give some examples, and we characterize geometrically our stan-

Kegelschnittbüschel und Inversion

The conjugacy mapping rel. to a complete quadrangle in a Pappian projective plane of characteristic ≠2 is constructed by using a bijection of the line set onto the bundle of conics through the diagonal points of the quadrangle. The inversion with center O of the inversion circle going through the point P in the Euclidean plane proves to be the product of the reflection at OP and the affine restriction of the conjugacy mapping rel. to the quadrangle having P as one of its vertices and O together with the circular points at infinity as diagonal points.

Octagonality conditions in projective and affine planes

A projective confined configuration ℭ with axis and centre will be introduced in terms of a non-degenerate octagon \(\mathfrak{D}\) satisfying some hypotheses on the position of its diagonal points (i.e. intersections of edges having distance 8 in the flag graph Γ(\(\mathfrak{D}\))) and its first minor diagonal lines (i.e. diagonal lines joining vertices of distance 6 in Γ(\(\mathfrak{D}\))). That confined configuration gives rise to a certain configurational condition whose affine specialization (i.e. the axis coincides with the line at infinity) is equivalent to the affine Pappos condition, whereas its ‘little’ specialization (i.e. the centre lies on the axis) turns out to be equivalent to the little Desargues condition. In Pappian projective planes ℭ can be completed to a configuration of type (124, 163).

The ternary rings of Desarguesian and Pappian planes ? A simple proof using perspectivities

The aim of this paper is to give a direct, simple proof of the well-known theorems — that the Hall ternary ring (R, T) of a Pappian projective plane is a linear ternary ring over a field, and that of a Desarguesian plane is a linear one over a skew field — by making repeated application of the perspectivity theorem in a Pappian plane and the characterization of Desarguesian planes in terms of perspectivities.

Collineations, Correlations, Polarities, and Conics

It is well known that planes of characteristic 2 behave differently from other Pappian projective planes. For this reason their detailed properties are usually ignored in books on synthetic projective geometry, especially when conies are being discussed. This can give rise to the misleading impression that planes of characteristic 2 are more difficult to deal with, while a cursory introduction to conies in such planes (e.g. Theorem 4.3) may suggest that the notion of “pole and polar” no longer exists.