Characterizing Combinatorial Geometries by Numerical Invariants

We show that the projective geometry PG(r − 1,q ) for r & 3 is the only rank- r(combinatorial) geometry with (q^r − 1) / (q − 1) points in which all lines have at least q + 1 points. For r = 3, these numerical invariants do not distinguish between projective planes of the same order, but they do distinguish projective planes from other rank-3 geometries. We give similar characterizations of affine geometries. In the core of the paper, we investigate the extent to which partition lattices and, more generally, Dowling lattices are characterized by similar information about their flats of small rank. We apply our results to characterizations of affine geometries, partition lattices, and Dowling lattices by Tutte polynomials, and to matroid reconstruction. In particular, we show that any matroid with the same Tutte polynomial as a Dowling lattice is a Dowling lattice.     

Extended dual affine planes

We study a class of diagram geometries, achieve a characterization of extended dual affine planes, and embed extended dual affine planes in extended projective planes. The geometries studied are rank 3 diagram geometries such that the residue of a point is a dual net, and the residue of a plane is linear; the dual of such a geometry has partitions on lines and planes which are reminiscent of parallelism of lines and planes of an affine 3-space. Examples of these geometries (some in dual form) include extended dual affine planes, Laguerre planes, 3-nets, and orthogonal arrays of strength 3. Theorem: Any such finite geometry satisfying Buekenhout's intersection property, and such that any two points are coplanar, is an extended dual affine plane (and has order 2, 4, or 10). Theorem: This geometry may be embedded in an extended projective plane of the same order.This research was partially supported by NSF Grant MCS-8102361.     

Totally skew embeddings of manifolds

We study a version of Whitney’s embedding problem in projective geometry: What is the smallest dimension of an affine space that can contain an n-dimensional submanifold without any pairs of parallel or intersecting tangent lines at distinct points? This problem is related to the generalized vector field problem, existence of non-singular bilinear maps, and the immersion problem for real projective spaces. We use these connections and other methods to obtain several specific and general bounds for the desired dimension.     

On regular {v,n}-arcs in finite projective spaces

A regular {v, n}-arc of a projective space P of order q is a set S of v points such that each line of P has exactly 0,1 or n points in common with S and such that there exists a line of P intersecting S in exactly n points. Our main results are as follows: (1) If P is a projective plane of order q and if S is a regular {v, n}-arc with n ≥ √q + 1, then S is a set of n collinear points, a Baer subplane, a unital, or a maximal arc. (2) If P is a projective space of order q and if S is a regular {v, n}-arc with n ≥ √q + 1 spanning a subspace U of dimension at least 3, then S is a Baer subspace of U, an affine space of order q in U, or S equals the point set Of U. © 1993 John Wiley & Sons, Inc.     

An extension of a theorem of G. Tallini

A Tallini set in a projective space P is a set Q of points of P such that each line not contained in Q intersects Q in at most two points. We prove that if P is a finite projective space with odd order q > 3 and dimension d > 2 and if |Q| > q^{d ? 1} + 2q^{d ? 3} + q^{d ? 4} + … + 1, then Q is essentially an orthogonal quadric. The proof of this theorem is based on a characterization of the orthogonal quadrics in every finite dimensional projective space (with possibly infinite order).     

Curious Characterizations of Projective and Affine Geometries

We prove the following characterization theorem: If any three of the following four matroid invariants—the number of points, the number of lines, the coefficient of λ^n − 2 in the characteristic polynomial, and the number of three-element dependent sets—of a rank-n combinatorial geometry (or simple matroid) are the same as those of a rank-n projective geometry, then it is a projective geometry (of the same order). To do this, we use a lemma which is of independent interest: If H is a geometry in which all the lines have exactly ℓ − 1 or ℓ points and G is a geometry with at least three of the four matroid invariants the same as H, then all the lines in G also have exactly ℓ − 1 or ℓ points. An analogue of the characterization theorem holds for affine geometries. Our methods also yield inequalities amongst the four matroid invariants.     

Ten Exceptional Geometries from Trivalent Distance Regular Graphs

The ten distance regular graphs of valency 3 and girth > 4 define ten non-isomorphic neighborhood geometries, amongst which a projective plane, a generalized quadrangle, two generalized hexagons, the tilde geometry, the Desargues configuration and the Pappus configuration. All these geometries are bislim, i.e., they have three points on each line and three lines through each point. We study properties of these geometries such as embedding rank, generating rank, representation in real spaces, alternative constructions. Our main result is a general construction method for homogeneous embeddings of flag transitive self-polar bislim geometries in real projective space.     

Regular Hjelmslev planes

A projective Hjelmslev plane is called regular iff it admits an Abelian collineation group that is regular on both the points and lines of the plane and that splits into a summand regular on the elements of any given neighborhood and another summand permuting the points and lines of the projective image plane regularly. Regular Hjelmslev planes are shown to correspond to so-called special difference sets. We construct regular Hjelmslev planes with parameters (qⁿ, q) for any prime power q and any natural number n as well as for infinitely many series of parameters (t, q), where t is not a power of q. Our construction also yields series of parameters for which the existence of a Hjelmslev plane was not known up to now as well as the first information on the existence of nontrivial collineations in the case of parameters (t, q) with t not a power of q.     

On (q 2 + q + 1)-sets of class [1, m,n]2 in PG(3, q)

A characterization of cones in PG(3, q) as sets of points of PG(3, q) of size q ² + q + 1 projecting from a point V a set of q + 1 points of a plane of PG(3, q) and with three intersection numbers with respect to the planes is given.     

Nonbinary quantum codes derived from finite geometries

The paper gives explicit parameters for several infinite families of q-ary quantum stabilizer codes. These codes are derived from combinatorial designs which arise from finite projective and affine geometries.     

A characterization of the minimalbasis of the torus

D. König asks the interesting question in [7] whether there are facts corresponding to the theorem of Kuratowski which apply to closed orientable or non-orientable surfaces of any genus. Since then this problem has been solved only for the projective plane ([2], [3], [8]). In order to demonstrate that König’s question can be affirmed we shall first prove, that every minimal graph of the minimal basis of all graphs which cannot be embedded into the orientable surface f of genusp has orientable genusp+1 and non-orientable genusq with 1≦q≦2p+2. Then let f be the torus. We shall derive a characterization of all minimal graphs of the minimal basis with the nonorientable genusq=1 which are not embeddable into the torus. There will be two very important graphs signed withX ₈ andX ₇ later. Furthermore 19 graphsG ₁,G ₂, ...,G ₁₉ of the minimal basisM(torus, >₄) will be specified. We shall prove that five of them have non-orientable genusq=1, ten of them have non-orientable genusq=2 and four of them non-orientable genusq=3. Then we shall point out a method of determining graphs of the minimal basisM(torus, >₄) which are embeddable into the projective plane. Using the possibilities of embedding into the projective plane the results of [2] and [3] are necessary. This method will be called saturation method. Using the minimal basisM(projective plane, >₄) of [3] we shall at last develop a method of determining all graphs ofM(torus, >₄) which have non-orientable genusq≧2. Applying this method we shall succeed in characterizing all minimal graphs which are not embeddable into the torus. The importance of the saturation method will be shown by determining another graphG ₂₀≠G ₁,G ₂, ...,G ₁₉ ofM(torus, >₄).     

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.     

Two-transitive groups on a hyperbolic unital

A classification given previously of all projective translation planes of order q² that admit a collineation group G admitting a two-transitive orbit of q+1 points is applied to show that the only projective translation planes of order q² admitting a hyperbolic unital acting two-transitively on a secant are the Desarguesian planes and the unital is a Buekenhout hyperbolic unital.     

Classes of optical orthogonal codes from arcs in root subspaces

We present new constructions for (n,w,λ) optical orthogonal codes (OOC) using techniques from finite projective geometry. In one case codewords correspond to (q-1)-arcs contained in Baer subspaces (and, in general, kth-root subspaces) of a projective space. In the other construction, we use sublines isomorphic to PG(2,q) lying in a projective plane isomorphic to PG(2,q^k), k>1. Our construction yields for each λ>1 an infinite family of OOCs which, in many cases, are asymptotically optimal with respect to the Johnson bound.     

Double k-sets in symplectic generalized quadrangles

In classical projective geometry, a double six of lines consists of 12 lines ? ₁, ? ₂, . . . , ? ₆, m ₁, m ₂, . . . , m ₆ such that the ? _i are pairwise skew, the m _i are pairwise skew, and ? _i meets m _j if and only if i ≠ j. In the 1960s Hirschfeld studied this configuration in finite projective spaces PG(3, q) showing they exist for almost all values of q, with a couple of exceptions when q is too small. We will be considering double-k sets in the symplectic geometry W(q), which is constructed from PG(3, q) using an alternating bilinear form. This geometry is an example of a generalized quadrangle, which means it has the nice property that if we take any line ? and any point P not on ?, then there is exactly one line through P meeting ?. We will discuss all of this in detail, including all of the basic definitions needed to understand the problem, and give a result classifying which values of k and q allow us to construct a double k-set of lines in W(q).     

Fluctuations in the number of points on smooth plane curves over finite fields

In this note, we study the fluctuations in the number of points on smooth projective plane curves over a finite field Fq as q is fixed and the genus varies. More precisely, we show that these fluctuations are predicted by a natural probabilistic model, in which the points of the projective plane impose independent conditions on the curve. The main tool we use is a geometric sieving process introduced by Poonen (2004) [8].     

Finite geometries which contain dual affine planes

Finite geometries in which each plane is projective or dual affine over the field of two elements, or affine over the field of three elements, are studied. It is shown that no connected geometry can mix all three species of planes, and the geometries in which projective and dual affine planes occur are classified.     

A unified construction for c*· c- and L · L*-geometries in projective spaces

We present two new constructions for c^* · c-geometries. The first provides, for each even prime powerq, a flag-transitive c^* · c-geometry of orderq–1 that is embedded in the projective space PG(3,q) and which is related with the Cameron-Fisher extended grids of odd type. The second construction is valid independently of the parity ofq. Forq even, it produces the same geometry as the first construction, and forq odd, two geometries related with some extended grids constructed by Meixner and Pasini.Next, by using some complementary models for c^* and L in a projective plane, we derive from our construction a new family of L · L^*-geometries embedded in PG(3,q). Forq even, these geometries are flag-transitive.     

Bilinear flocks

A flock in PG(3, q) is a set of q planes which do not contain the vertex of a cone and have the property that the intersections of the planes of the flock with the cone partition the points of the cone except for the vertex. In this paper, we examine flocks, called bilinear flocks, where the planes of the flock pass through at least one of two distinct lines, called supporting lines in PG(3, q). We classify and provide examples of cones that admit bilinear flocks whose supporting lines intersect in PG(3, q). We also examine bilinear flocks whose supporting lines are skew, providing an example and also showing that this situation can not occur under certain conditions.