Die Invarianten Einer Klasse Projektiver Hjelmslev-Ebenen

Associated with every finite PH-plane (projective Hjelmslev plane)H is a pair of integer invariants (t,r): r denotes the order of the projective plane paired withH, t² the number of points in each neighborhood ofH. A pair (t,r) of positive integers is called a Lenz-pair if there exist orders r=q₂,...,q_n of projective planes such that t=q₂q₃...q_n and, for each i>2, either (a) q_i+1=q_i or (b) q_{i + 1} = (q₂q₃...q_i)(r + 1)–1 holds. A special Lenz-pair is a Lenz-pair for which every q_i is a prime power. Our major result asserts that the invariants of a finite, regular, minimally uniform PH-plane are always a Lenz-pair. In the converse direction, we prove that every special Lenz-pair may be realized as the invariant pair of some finite, regular, minimally uniform PH-plane.

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Ich danke der Alexander von Humboldt Stiftung für ihre Unterstützung während der Vorbereitung dieser Arbeit. Auch danke ich der University of Florida für die partielle Unterstützung durch ein Faculty Development Grant. Der Technischen Hochschule Darmstadt gilt mein Dank für die erwiesene Gastfreundschaft.     

Orthogonal resolutions of lines in AG(n,q)

A resolution of the lines of AG(n,q) is a partition of the lines classes (called resolution classes) such that every point of the geometry is on exactly one line of each resolution class. Two resolutions R,R' of AG(n,q) are orthogonal if any resolution class from R has at most one line in common with any class from R'. In this paper, we construct orthogonal resolutions on AG(n,q) for all n=2ⁱ⁺¹, i=1,2,…, and all q>2 a prime power. The method involves constructing AG(n,q) from a finite projective plane of order q^n-1 and using the structure of the plane to display the orthogonal resolutions.     

New integer pairs for Hjelmslev planes

Associated with every finite projective Hjelmslev plane is an invariant pair(t, r); t is the order of the Hjelmslev plane andr is the order of the underlying projective plane. The aim of this paper is to give some new constructions of Hjelmslev planes with an invariant pair (t, 2). First we construct a PH-plane with the invariant pair (20, 2). Using this, 16 more invariant pairs (t, 2) witht 1000 are obtained. In all, we thus obtain 17 new PH-planes with invariant pairs (t, 2),t 1000.     

On the theorems of Drake and Lenz

A generalization and an improvement of the results of Drake and Lenz on the constructions of projective Hjelmslev planes are obtained. Using this, some new series of invariant pairs including 4 new pairs (t, 2),t 1000, for projective Hjelmslev planes are obtained.     

Minimal non-orientable matroids in a projective plane

We construct a new family of minimal non-orientable matroids of rank three. Some of these matroids embed in Desarguesian projective planes. This answers a question of Ziegler: for every prime power q, find a minimal non-orientable submatroid of the projective plane over the q-element field.     

The Classification of Projective Planes of Order q3 with Cone-Representations in PG (6, q)

The classification of cone-representations of projective planes of orderq ³ of index 3 and rank 4 (and so in PG(6,^q)) is completed. Any projective plane with a non-spread representation (being a cone-representation of the second kind) is a dual generalised Desarguesian translation plane, as found by Jha and Johnson, and conversely. Indeed, given any collineation of PG(2,^q) with no fixed points, there exists such a projective plane of order q³ , where q is a prime power, that has the second kind of cone-representation of index 3 and rank 4 in PG(6,q). An associated semifield plane of order q ³ is also constructed at most points of the plane. Although Jha and Johnson found this plane before, here we can show directly the geometrical connection between these two kinds of planes.     

Recent results on designs with classical parameters

We report on recent results concerning designs with the same parameters as the classical geometric designs PG _d (n, q) formed by the points and d-dimensional subspaces of the n-dimensional projective space PG(n, q) over the field GF(q) with q elements, where 1 ???d ???n?1. The corresponding case of designs with the same parameters as the classical geometric designs AG _d (n, q) formed by the points and d-dimensional subspaces of the n-dimensional affine space AG(n, q) will also be discussed, albeit in less detail.     

Fano subplanes in finite Figueroa planes

It has been conjectured that all non-desarguesian projective planes contain a Fano subplane. The Figueroa planes are a family of non-translation planes that are defined for both infinite orders and finite order q ³ for q > 2 a prime power. We will show that there is an embedded Fano subplane in the Figueroa plane of order q ³ for q any prime power.     

PBIB designs with m associate classes induced by full rank matrices modulo pm

This paper presents series of PBIB designs with m associate classes in which the treatment set is a subset of the Z(p^m)-module of n × 1 vectors over the ring of integers modulo p^m, p any prime. The association scheme of this series of designs is determined by the Fuller canonical form under row equivalence of n × 2 matrices [a,b] for vectors a and b in the treatment set. The blocking procedure utilizes full rank s × n matrices over Z(p^m), 1 ? s ? n ? 2, n ? 3. For m = 2, n = 3, s =1 and for each prime p, each PBIB is regular divisible and yields a finite proper uniform projective Hjelmslev plane with parameters j = p and k = p(p + 1).     

Embeddability of restricted linear spaces

In this note we examine the problem of embedding into finite projective planes finite linear spaces with p points and q lines satisfying (q ? p)² ? p and q ? 2.     

Proof of a conjecture of Segre and Bartocci on monomial hyperovals in projective planes

The existence of certain monomial hyperovals D(x ^k ) in the finite Desarguesian projective plane PG(2, q), q even, is related to the existence of points on certain projective plane curves g _k (x, y, z). Segre showed that some values of k (k?=?6 and 2 ⁱ ) give rise to hyperovals in PG(2, q) for infinitely many q. Segre and Bartocci conjectured that these are the only values of k with this property. We prove this conjecture through the absolute irreducibility of the curves g _k .     

Order and topology in projective Hjelmslev planes

The concept of an ordered projective Hjelmslev plane was intuitively introduced by Hjelmslev in Einleitung in die allgemeine Kongruenglehre ([9], [10]).This paper is concerned with formalizing and examing preorderings and orderings for projective Hjelmslev planes. In addition we show that orderings generated topologies of the point and line sets which render the plane a topological Hjelmslev plane ([19], [13]). These planes — unlike the ordinary ordered planes ([18]) — are, due to the existence of infinitesimals, non-archimedian, non-compact and disconnected with the neighbour classes as certain quasi-components.The authors gratefully acknowledge the support of the Natural Sciences and Engineering Research Council of Canada.     

Fixed points of projectivities of prime order

This work begins with a review of the classical results for fixed points of projectivities in a projective plane over a general commutative field. The second section of this work features all the material necessary to prove the main result, which is presented in Theorem 2.8. It is shown that, in a finite projective plane of order q, there exists a projectivity g? of prime order p?>?3 if and only if p divides exactly one of the integers q ? 1, q, q?+?1, q ² + q + 1. Theorem 2.8 establishes a correspondence between the possible structures of points fixed by g?, as presented in Theorem 1.3, and the integer that is divisible by p. The special case of p = 2 is handled in Sect. 2.1, where it is shown that every involution is a harmonic homology for q odd and an elation for q even. The special case of p?=?3 is handled in Sect. 2.2, and Theorem 2.8 is adapted for p?=?3 and presented as Theorem 2.15. An application of Theorems 2.8 and 2.15 is determining the sizes of (n, r)-arcs that are stabilized by projectivities of prime order p in the finite projective plane of order q; in Sect. 3, this application is presented in Propositions 3.2 and 3.3.     

Hjelmslevgruppen mit Nachbarhomomorphismus

Hjelmslev groups have been introduced by F. Bachmann ([1], [2]) in order to study plane metric geometries in a general sense: For example two points may have none or two lines joining them. Let (G,S) and (-G,¯ S) be Hjelmslev groups and let be a Hjelmslev homomorphism from (G,S) onto (¯G, ¯S). It is shown that — under certain assumptions — the group plane of (G, S) can be embedded into the projective Hjelmslev plane over a local ringR and thatG is isomorphic to a subgroup of an orthogonal group O ₃ ⁺ (V,f). The result may be considered as a generalization of the main theorem in Bachmann [1].     

Polarities in n-uniform projective Hjelmslev planes

In [9] the author has studied polarities in finite 2-uniform projective Hjelmslev planes. The present paper deals with polarities in finite n-uniform projective Hjelmslev planes (n 2).The author's research was supported by IWONL grant no. 840037.     

Switching Sets Regolari E Cubiche Rigate Di P G (5, q ) Piani Di Tralsazione Ad Essi Associati G (5, q ) Piani Di Tralsazione Ad Essi Associati

In a recent paper, the authors studied some algebraic hypersurfaces of the third order in the projective spacePG(5,q) and they called them ruled cubics, since they possess three systems of planes. Any two of these constitute a regular switching set and furthermore, if Σ is a given regular spread ofPG(5,q), one of the three systems is contained in Σ. The subject of this note is to prove, conversely, that every regular switching set (Φ, Φ′) with Φ ⊂ Σ is a ruled cubic and to construct, for a generic choice of the projective reference system inP G(5,q), the quasifield which coordinatizes the translation plane Π associated with the spread (Σ − Φ) ∪ Φ′. The planes Π, of orderq ³, are a generalization of the finite Hall planes.     

Mixed partitions and related designs

We define a mixed partition of Π =  PG(d, q ^r ) to be a partition of the points of Π into subspaces of two distinct types; for instance, a partition of PG(2n ? 1, q ²) into (n ? 1)-spaces and Baer subspaces of dimension 2n ? 1. In this paper, we provide a group theoretic method for constructing a robust class of such partitions. It is known that a mixed partition of PG(2n ? 1, q ²) can be used to construct a (2n ? 1)-spread of PG(4n ? 1, q) and, hence, a translation plane of order q ²ⁿ . Here we show that our partitions can be used to construct generalized Andrè planes, thereby providing a geometric representation of an infinite family of generalized Andrè planes. The results are then extended to produce mixed partitions of PG(rn ? 1, q ^r ) for r ≥ 3, which lift to (rn ? 1)-spreads of PG(r ² n ? 1, q) and hence produce $2-(q^{r^2n},q^{rn},1)$ (translation) designs with parallelism. These designs are not isomorphic to the designs obtained from the points and lines of AG(r, q ^rn ).