Simplectic spreads and finite semifields

It is well known that associated with a translation plane π there is a family of equivalent spreads. In this paper, we prove that if one of these spreads is symplectic and π is finite, then all the associated spreads are symplectic. Also, using the geometric intepretation of the Knuth’s cubical array, we prove that a symplectic semifield spread of dimension n over its left nucleus is associated via a Knuth operation to a commutative semifield of dimension n over its middle nucleus.      

A characterization of reguli in PG(3,K) and PG(5,K)

In this note, a regulus of lines in PG(3,K) and a regulus of planes in PG(5,K) are characterized by incidence properties.     

On a class of finite linear spaces with few lines

A famous result of de Bruijn and Erdős (Indag. Math. 10 (1948) 421–423) states that a finite linear space has at least as many lines as points, with equality only if it is a projective plane or a near-pencil. This result led to the problem of characterizing finite linear spaces for which the difference between the number b of lines and the number v of points is assigned.

In this paper finite linear spaces with b−vm, m being the minimum number of lines on a point, are characterized.     

On the Ideals of Secant Varieties of Segre Varieties

We establish basic techniques for determining the ideals of secant varieties of Segre varieties.We solve a conjecture of Garcia, Stillman, and Sturmfels on the generators of the ideal of the first secant variety in the case of three factors and solve the conjecture set-theoretically for an arbitrary number of factors. We determine the low degree components of the ideals of secant varieties of small dimension in a few cases.     

On the Duals of Segre Varieties

The reflexivity, the (semi-)ordinariness, the dimension of dual varieties and the structure of Gauss maps are discussed for Segre varieties, where a Segre variety is the image of the product of two or more projective spaces under Segre embedding. A generalization is given to a theorem of A. Hefez and A. Thorup on Segre varieties of two projective spaces. In particular, a new proof is given to a theorem of F. Knop, G. Menzel, I. M. Gelfand, M.M. Kapranov and A. V. Zelevinsky that states a necessary and sufficient condition for Segre varieties to have codimension one duals. On the other hand, a negative answer is given to a problem raised by S. Kleiman and R. Piene as follows: For a projective variety of dimension at least two, do the Gauss map and the natural projection from the conormal variety to the dual variety have the same inseparable degree?     

On two classes of finite supersoluble groups

Let ? be a complete set of Sylow subgroups of a finite group G, that is, a set composed of a Sylow p-subgroup of G for each p dividing the order of G. A subgroup H of G is called ?-S-semipermutable if H permutes with every Sylow p-subgroup of G in ? for all p?π(H); H is said to be ?-S-seminormal if it is normalized by every Sylow p-subgroup of G in ? for all p?π(H). The main aim of this paper is to characterize the ?-MS-groups, or groups G in which the maximal subgroups of every Sylow subgroup in ? are ?-S-semipermutable in G and the ?-MSN-groups, or groups in which the maximal subgroups of every Sylow subgroup in ? are ?-S-seminormal in G.     

Symplectic semifield planes and -linear codes

There are lovely connections between certain characteristic 2 semifields and their associated translation planes and orthogonal spreads on the one hand, and -linear Kerdock and Preparata codes on the other. These inter-relationships lead to the construction of large numbers of objects of each type. In the geometric context we construct and study large numbers of nonisomorphic affine planes coordinatized by semifields; or, equivalently, large numbers of non-isotopic semifields: their numbers are not bounded above by any polynomial in the order of the plane. In the coding theory context we construct and study large numbers of -linear Kerdock and Preparata codes. All of these are obtained using large numbers of orthogonal spreads of orthogonal spaces of maximal Witt index over finite fields of characteristic 2.

We also obtain large numbers of ``boring' affine planes in the sense that the full collineation group fixes the line at infinity pointwise, as well as large numbers of Kerdock codes ``boring' in the sense that each has as small an automorphism group as possible.

The connection with affine planes is a crucial tool used to prove inequivalence theorems concerning the orthogonal spreads and associated codes, and also to determine their full automorphism groups.

     

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.     

Ovoids of Parabolic Spaces

A new ovoid in the orthogonal space O(5,3⁵) is presented, along with its associated spreads and (semifield) translation planes. Sundry results on ovoids and spreads are given. In particular, we complete the calculation of the stabilisers of the known O(5,q) ovoids.     

On the Non-Splitting of the Normal Bundle Sequence

In a recent article, Paltin Ionescu and Flavia Repetto proved that if X ? ? ⁿ is a smooth projective variety over ? such that its normal bundle sequence splits over some curve C ? X, then X a linear subspace in ? ⁿ . In this note, we give a purely geometric proof of this result that is valid in arbitrary characteristic.     

The covering radius of PGL(3,q)

In this note, with a purely geometric approach, the covering radius of the group $\mathrm{PGL}(3,q)$ is determined. Also, a new proof establishing the covering radii of $\mathrm{PGL}(2,q)$ and $\mathrm{AGL}(1,q)$ is provided.     

A lower bound for the minimum weight of the dual 7-ary code of a projective plane of order 49

Existing bounds on the minimum weight d ^⊥ of the dual 7-ary code of a projective plane of order 49 show that this must be in the range 76 ≤ d ^⊥ ≤ 98. We use combinatorial arguments to improve this range to 88 ≤ d ^⊥ ≤ 98, noting that the upper bound can be taken to be 91 if the plane has a Baer subplane, as in the desarguesian case. A brief survey of known results for the minimum weight of the dual codes of finite projective planes is also included. Dedicated to Dan Hughes on the occasion of his 80th birthday.     

On the Ree Unital

The embedding of a Ree Unital in a finite projective plane Π of order up to q ⁴ is investigated when the Ree group is induced on by a collineation group of Π. In particular, it is shown that such a embedding is not admissible for q ≠ 3, extending in this way a result of Lüneburg dating back to 1965.