On parallelisms in finite projective spaces

Geometriae Dedicata -     

Partial parallelisms in finite projective spaces

We show the existence of a parallelism of P–U, where P is a finite projective space and U is a subspace of P with dim P–dim U=2 ⁱ . As a consequence we prove a lower bound for the maximum number of disjoint spreads of P.To Helmut Salzmann on the occasion of his 60th birthday     

Linear sets in finite projective spaces

In this paper linear sets of finite projective spaces are studied and the “dual” of a linear set is introduced. Also, some applications of the theory of linear sets are investigated: blocking sets in Desarguesian planes, maximum scattered linear sets, translation ovoids of the Cayley Hexagon, translation ovoids of orthogonal polar spaces and finite semifields. Besides “old” results, new ones are proven and some open questions are discussed.     

On t-covers in finite projective spaces

A t-cover of the finite projective space PG(d,q) is a setS of t-dimensional subspaces such that any point of PG(d,q) is contained in at least one element ofS. In Theorem 1 a lower bound for the cardinality of a t-coverS in PG(d,q) is obtained and in Theorem 2 it is shown that this bound is best possible for all positive integers t,d and for any prime-power q.     

Baer cones in finite projective spaces

Summary Let R and V be two skew subspaces with dimensions r and v of P=PG(d,q). If q is a square, then there is a Baer subspace V^* of V, i.e. a subspace of dimension v and order q. We call the set C(R,V^*)= , where the union is taken over all PV^*, aBaer cone oftype (r,v).A (t.s)- blocking set is a set B of points of p such that any (d-t)-dimensional subspace is incident with at least one point of B, and no sdimensional subspace is contained in B. We show that for every (t,s)-blocking set B in PG(d,q) we have¦B≥ q^t +...+1 + q(q^t–1+...+q^s–1) with equality if and only if B is a Baer cone of type (s–2,2(t–s+1)).     

Generalized Veronesean embeddings of projective spaces

We classify all embeddings θ: PG(n, q) → PG(d, q), with $d geqslant tfrac{{n(n + 3)}} {2}$d geqslant tfrac{{n(n + 3)}} {2}, such that θ maps the set of points of each line to a set of coplanar points and such that the image of θ generates PG(d, q). It turns out that d = ?n(n+3) and all examples are related to the quadric Veronesean of PG(n, q) in PG(d, q) and its projections from subspaces of PG(d, q) generated by sub-Veroneseans (the point sets corresponding to subspaces of PG(n, q)). With an additional condition we generalize this result to the infinite case as well.     

On multiple caps in finite projective spaces

In this paper, we consider new results on (k, n)-caps with n > 2. We provide a lower bound on the size of such caps. Furthermore, we generalize two product constructions for (k, 2)-caps to caps with larger n. We give explicit constructions for good caps with small n. In particular, we determine the largest size of a (k, 3)-cap in PG(3, 5), which turns out to be 44. The results on caps in PG(3, 5) provide a solution to four of the eight open instances of the main coding theory problem for q = 5 and k = 4.     

Caps in finite projective spaces of odd order

In this paper upper bounds on the number of points of large complete caps in PG(n, q), (q odd,n3) are derived. The previously known bounds of Hirschfeld and Segre are improved using recent bounds on the cardinality of the second largest complete cap in the plane PG(2,q).Research of this author is supported by OTKA Grants Nos. 2505 and T-014105     

On smallest covers of finite projective spaces

A t-cover of a finite projective space ℙ is a set of t-dimensional subspaces covering all points of ℙ. Beutelspacher [1] constructed examples of t-covers and proved that his examples are of minimal cardinality. We shall show that all examples of minimal cardinality “look like” the examples of Beutelspacher.     

Blockades and baer subspaces in finite projective spaces

Let P=PG(2t + 1, q) denote the projective space of order q and of dimension 2t+13. A set of lines of P is called a blockade if it fulfills the following two conditions.

Pasch geometric spaces inducing finite projective planes

Summary A geometric space over a geometric sfield of dimension three induces a protective plane. A relation between the order of the projective plane and that of the geometric sfield is obtained. For a particular order of the sfield, the induced projective plane is shown to be desarguesian.     

Line-transitive collineation groups of finite projective spaces

A collineation group Γ ofPG(d, q), d ≧ 3, which is transitive on lines is shown to be 2-transitive on points unlessd=4,q=2 and |Γ|=31·5. Research supported in part by NSF Grant GP 28420.