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.     

Symmetric Gobos in a finite projective plane

A gobo G in any incidence structure K is a (perhaps degenerate) tactical configuration having the property that no three points in G are collinear and no three lines in G are concurrent. General results are obtained where K is a finite projective plane of order n and G has k points and k lines such that each point (line) lies on r lines (points) of G. Particular attention is called to the contrast between the case r = 1 and the case r ≠ 1 when k = n.     

New upper bounds on the minimum size of covering designs

Let D = {B₁, B₂,…, B_b} be a finite family of k-subsets (called blocks ) of a v-set X(v) = {1, 2,…, v} (with elements called points ). Then D is a (v, k, t) covering design or covering if every t-subset of X(v) is contained in at least one block of D. The number of blocks, b, is the size of the covering, and the minimum size of the covering is called the covering number , denoted C(v, k, t). This article is concerned with new constructions of coverings. The constructions improve many upper bounds on the covering number C(v, k, t) © 1998 John Wiley & Sons, Inc. J Combin Designs 6:21–41, 1998     

Curvature-free upper bounds for the smallest area of a minimal surface

In this paper we will present two upper estimates for the smallest area of a possibly singular minimal surface in a closed Riemannian manifold Mⁿ with a trivial first homology group. The first upper bound will be in terms of the diameter of Mⁿ, the second estimate will be in terms of the filling radius of a manifold, leading also to the estimate in terms of the volume of Mⁿ. If n = 3 our upper bounds are for the smallest area of a smooth embedded minimal surface. After that we will establish similar upper bounds for the smallest volume of a stationary k-dimensional integral varifold in a closed Riemannian manifold Mⁿ with . The above results are the first results of such nature. Received: October 2004 Revision: May 2005 Accepted: June 2005     

On subplane partitions of a finite projective plane

Bruck (1960) notes that every finite Desarguesian plane of order m² can be partitioned into disjoint subplanes of order m. His method follows directly from the discovery by Singer (1938) that every such plane is cyclic and may be derived from a difference set. The present paper begins investigating the existence of other partitions which do not follow naturally from the difference set representation of the plane. The procedure is based on the idea, defined herein, of conjugacy with respect to a given subplane. When m = 2, only Bruck's partition is possible, but new results have been obtained in the cases m = 3, 4, 5, 7. This paper treats the case m = 3 in some detail.     

Characterization of seminuclear sets in a finite projective plane

LetC be a set ofq + a points in the desarguesian projective plane of orderq, such that each point ofC is on exactly 1 tangent, and onea+ 1-secant (a>1). Then eitherq=a + 2 andC consists of the symmetric difference of two lines, with one further point removed from each line, orq=2a + 3 andC is projectively equivalent to the set of points {(0,1,s),(s, 0, 1),(1,s, 0): -s is not a square inGF(q)}.     

On the nuclei of a pointset of a finite projective plane

In this paper we study subset N(K) of a set K the points of which are not in a collinear triplet of K and prove that ¦N(K)¦(q+1)/2 or N(K)=K if K is a (q+1)-set of PG(2,q). We describe all the k-arcs of AG(2,q) the secants of which meet the ideal line exactly in k points.Dedicated to Professor Ferenc Kárteszi on his 80^th birthday     

1-Embeddability of complete multipartite graphs on the projective plane

A graph is 1-embeddable on a closed surface if there exists a drawing of the graph on the surface such that each edge crosses at most one other edge at a point. In this paper, we determine all the 1-embeddable complete k-partite graphs on the projective plane.     

Curves containing all points of a finite projective Galois plane

In the projective plane $PG(2,q)$ over a finite field of order q, a Tallini curve is a plane irreducible (algebraic) curve of (minimum) degree $q+2$ containing all points of $PG(2,q)$. Such curves were investigated by G. Tallini [8], [9] in 1961, and by Homma and Kim [5] in 2013. Our results concern the automorphism groups, the Weierstrass semigroups, the Hasse–Witt invariants, and quotient curves of the Tallini curves.     

The splitting number of the complete graph in the projective plane

If a given graphG can be obtained bys vertex identifications from a suitable graph embeddable in the projective plane ands is the minimum number for which this is possible thens is called the splitting number ofG in the projective plane. Here a formula for the splitting number of the complete graph in the projective plane is derived.     

Rigidity of certain finite group actions on the complex projective plane

Partially supported by NSERC grant A4000