Pia Maria Lo Re

On regular planar spaces of type (k,n)

In this paper, we study finite regular planar spaces (S,L,P) of type (k,n) whose planes pairwise have non-empty intersection. We show that |L|>=|P| and if |L|=|P|, then (S,L,P) is PG(4,n); if |L|>|P| and n=<30000, then (S,L,P) is PG(3,n).

On finite {1,2,3}-semiaffine planes

In this paper, we investigate {1,2,3}-semiaffine planes. All such planes of order n >51 shall be classified. It turns out that they are embeddable into projective planes of the same order n in the most natural way.

A combinatorial characterization of the Klein quadric

We give a combinatorial characterization of the Klein quadric in terms of its incidence structure of points and lines. As an application, we obtain a combinatorial proof of a result of Havlicek.

Embedding the planar structure of 4-dimensional linear spaces into projective spaces

It is known that a linear spaces of dimensiond has at least as many hyperplanes as points with equality if it is a (possibly degenerate) projective space. If there are only a few more hyperplanes than points, then the linear space can still be embedded in a projective space of the same dimension. But even if the difference between the number of hyperplanes and points is too big to ensure an embedding, it seems likely that the linear space is closely related to a projective space. We shall demonstrate this in the cased=4.

Generalized Q -sets in a finite locally projective planar space

Abstract Generalized Q-sets in a locally projective planar space S of order n are defined. If S contains a non-degenerate generalized Q-set Q and the order n (>4) of S is even, we prove that S is embeddable in PG(3,n) and Q is isomorphic to an ovoid or a hyperbolic quadric of PG(3,n).

A characterization of linear spaces based on the number of transversals

We characterize a class of linear spaces by the property that through any point outside two disjoint, but non-parallel lines there is at most one transversal.

On some classes of linear spaces embedded in a Pappian plane

Abstract We deal with the following problem. Let L be a suitable finite linear space embedded in a Pappian plane P and suppose that L is embeddable in a finite projective plane π′ of order n . It is true that a finite subplane π of P isomorphic to π′ containing L exists?