A characterisation of spreads ovally-derived from Desarguesian spreads

We characterise all spreads that are obtainable from Desarguesian spreads by replacing a partial spread consisting of translation ovals; the corresponding ovally-derived planes are generalised André planes, of order 2 ^N , although not all generalised André planes are ovallyderived from Desarguesian planes. Our analysis allows us to obtain a complete classification of all nearfield planes that are ovally-derived from Desarguesian planes. It turns out that whether or not a nearfield plane is ovally-derived from a Desarguesian plane depends solely on the parametersq andr, where GF (q) is the kern, andr is the dimension of the plane. Our results also imply that a Hall plane of even orderq ² can be ovally-derived from a Desarguesian spread if and only ifq is a square.     

Desarguesian spreads

We review some constructions related with desarguesian spreads of projective spaces of finite dimension over a skewfield.     

Linear sets from projection of Desarguesian spreads

Every linear set in a projective space is the projection of a subgeometry, and most known characterizations of linear sets are given under this point of view. For instance, scattered linear sets of pseudoregulus type are obtained by considering a Desarguesian spread of a subgeometry and projecting from a vertex which is spanned by all but two director spaces. In this paper we introduce the concept of linear sets of h-pseudoregulus type, which turns out to be projected from the span of an arbitrary number of director spaces of a Desarguesian spread of a subgeometry. Among these linear sets, we characterize those which are h-scattered and solve the equivalence problem between them; a key role is played by an algebraic tool recently introduced in the literature and known as Moore exponent set. As a byproduct, we classify asymptotically h-scattered linear sets of h-pseudoregulus type.     

A geometric characterisation of linear k-blocking sets

In this paper we prove that any linear k-blocking set is either a canonical subgeometry or a projection of some canonical subgeometry. Received 6 February 2001.     

Locally Desarguesian spaces

In Riemannian spaces, locally Desarguesian spaces have constant curvature and are therefore locally symmetric. This does not hold for Finsler spaces, so that locally Desarguesian spaces represent a generalization other than the obvious one we studied previously of (certain) Riemannian symmetric spaces. In this paper we discuss them in detail; as an example of the results obtained we mention that a simply connected locally Desarguesian space without conjugate points is globally Desarguesian. Applications are then given to spaces which are locally symmetric in a wider sense. We also study (and in Minkowski spaces determine exactly) the properties of functions which measure the distance of a point from those on a line.     

Quasi-ordered Desarguesian affine spaces

Starting from Euclidean spaces with a betweenness function, which is compatible with the congruence relation, we introduce quasi-orderings as a common generalization of half-orderings and semi-orderings. Quasi-ordered Desarguesian affine spaces are described algebraically by guasi-ordered skew fields.Dedicated to Professor Dr. W. Benz on the occasion of his 60 th birthday     

Desarguesian partial parallelisms

This article establishes connections between Desarguesian partial parallelisms of deficiency one in PG(3,q) and translation planes of orderq ⁴ admitting a collineation group isomorphic to SL(2,q) which is generated by Baer collineations.The author is grateful to Professor Alan Prince for helpful conversations with respect to this article and, in particular, with respect to the actual bounds for certain maximal partial spreads.The author is grateful to the referee for helpful comments in the writing of this article.     

Desarguesian nets without ovals

Let Π be the Desarguesian affine plane coordinatized by a division ring D with charD≠2. For n=6 and also for n=7, we prove that Π holds an n-net without an oval if and only if |D|?9.     

Almost Desarguesian Maximal Partial Spreads

Constructions are given of various classesof maximal partial spreads in PG(3,2 ^r ) whose partialspreads consist of q/2 reguli sharing a line. Further,characterization results are given for the main classes of constructedmaximal partial spreads.     

A remark on symplectic spreads

We first note that each element of a symplectic spread of PG(2n − 1, 2 ^r ) either intersects a suitable nonsingular quadric in a subspace of dimension n − 2 or is contained in it, then we prove that this property characterises symplectic spreads of PG(2n − 1, 2 ^r ). As an application, we show that a translation plane of order q ⁿ , q even, with kernel containing GF(q), is defined by a symplectic spread if and only if it contains a maximal arc of the type constructed by Thas (Europ J Combin 1:189–192, 1980).     

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).     

Constructions of unitals in Desarguesian planes

We present a new construction of non-classical unitals from a classical unital U in . The resulting non-classical unitals are B-M unitals. The idea is to find a non-standard model Π of with the following three properties:

(i)

points of Π are those of ;

(ii)

lines of Π are certain lines and conics of ;

(iii)

the points in U form a non-classical B-M unital in Π.

Our construction also works for the B-T unital, provided that conics are replaced by certain algebraic curves of higher degree.