On the classification of semifield flocks

A classical lemma of Weil is used to characterise quadratic polynomials f with coefficients GF(qⁿ), q odd, with the property that f(x) is a non-zero square for all x∈GF(q). This characterisation is used to prove the main theorem which states that there are no subplanes of order q contained in the set of internal points of a conic in PG(2,qⁿ) for q?4n²−8n+2. As a corollary to this theorem it then follows that the only semifield flocks of the quadratic cone of PG(3,qⁿ) for those q exceeding this bound are the linear flocks and the Kantor-Knuth semifield flocks.     

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.     

The six semifield planes associated with a semifield flock

In 1965 Knuth (J. Algebra 2 (1965) 182) noticed that a finite semifield was determined by a 3-cube array (a_ijk) and that any permutation of the indices would give another semifield. In this article we explain the geometrical significance of these permutations. It is known that a pair of functions (f,g) where f and g are functions from GF(q) to GF(q) with the property that f and g are linear over some subfield and g(x)²+4xf(x) is a non-square for all x∈GF(q)∗, q odd, give rise to certain semifields, one of which is commutative of rank 2 over its middle nucleus, one of which arises from a semifield flock of the quadratic cone, and another that comes from a translation ovoid of Q(4,q). We show that there are in fact six non-isotopic semifields that can be constructed from such a pair of functions, which will give rise to six non-isomorphic semifield planes, unless (f,g) are of linear type or of Dickson-Kantor-Knuth type. These six semifields fall into two sets of three semifields related by Knuth operations.     

Flocks and Partial Flocks of Hyperbolic Quadrics via Root Systems

We construct three infinite families of partial flocks of sizes 12, 24 and 60 of the hyperbolic quadric of PG(3, q), for q congruent to -1 modulo 12, 24, 60 respectively, from the root systems of type D ₄, F ₄, H ₄, respectively. The smallest member of each of these families is an exceptional flock. We then characterise these partial flocks in terms of the rectangle condition of Benz and by not being subflocks of linear flocks or of Thas flocks. We also give an alternative characterisation in terms of admitting a regular group fixing all the lines of one of the reguli of the hyperbolic quadric.     

Conical flocks,partial flocks,derivation, and generalized quadrangles

L. Bader, G. Lunardon and J. A. Thas have shown that a flock ₀ of a quadratic cone in PG(3, q), q odd, determines a set ={⁰,₁,...,_q} of q+1 flocks. Each _j , 1jq, is said to be derived from ₀. We show that, by derivation, the flocks with q=3 ^e arising from the Ganley planes yield an inequivalent flock for q27. Further, we prove that the Fisher flocks (q odd, q5) are the unique nonlinear flocks for which (q–1)/2 planes of the flock contain a common line. This result is used to show that each of the flocks derived from a Fisher flock is again a Fisher flock. Finally, we prove that any set of q–1 pairwise disjoint nonsingular conics of a cone can be extended to a flock. All these results have implications for the theory of translation planes.     

Extending partial flocks containing linear subflocks

It is shown that a partial flock of a quadratic cone in PG(3,q) which properly contains a linear subflock of at least (q–1)/2 conics may be uniquely extended to a linear or Fisher flock.Dedicated to Professor T. G. Ostrom, on the occasion of his eightieth birthday     

A Geometric Construction of the Hyperbolic Fibrations Associated with a Flock, q Even

Baker et al. [2] show, via algebraic methods, that a regular hyperbolic fibration of PG(3, q) with constant back half gives rise to a flock of a quadratic cone in PG(3, q), and conversely. In this paper a geometric construction for q even of the flock from the hyperbolic fibration, and conversely, will be described. A proof will be given that this geometric construction indeed corresponds to the known algebraic one. Deirdre Luyckx - The author is Postdoctoral Fellow of the Fund for Scientific Research—Flanders (Belgium) (F.W.O.—Vlaanderen).     

Flocks of a quadratic cone in PG(3, q), q ≤ 8

If (O) is a quadratic cone in PG(3,q), with vertex x, then a flock of (O) is a partition of (O)-{x} into q disjoint conics. With such a flock there correspond a translation plane of order q ² and a generalized quadrangle of order (q ², q). Here we determine all flocks of (O) for q 8.     

Characterisations of Flock Quadrangles

We characterise the Hermitian and Kantor flock generalized quadrangles of order (q ²,q), q even, (associated with the linear and Fisher–Thas–Walker flocks of a quadratic cone, and the Desarguesian and Betten–Walker translation planes) in terms of a self-dual subquadrangle. Equivalently, we show that a herd which contains a translation oval must be associated with the linear or Fisher–Thas–Walker flock. This result is a consequence of the determination of all functions which satisfy a certain absolute trace equation whose form is remarkably similar to that of an equation arising in recent studies of ovoids in three-dimensional projective space of finite order q.     

Triads, Flocks of Conics and Q -(5,q)

We show that if an ovoid of Q (4,q),q even, admits a flock of conics then that flock must be linear. It follows that an ovoid of PG (3,q),q even, which admits a flock of conics must be an elliptic quadric. This latter result is used to give a characterisation of the classical example Q ^-(5,q) among the generalized quadrangles T ₃( ), where is an ovoid of PG (3q) and q is even, in terms of the geometric configuration of the centres of certain triads.     

A condition for the existence of ovals in PG(2, q), q even

A condition is found that determines whether a polynomial over GF(q) gives an oval in PG(2, q), q even. This shows that the set of all ovals of PG(2, q) corresponds to a certain variety of points of PG((q–4)/2, q). The condition improves upon that of Segre and Bartocci, who proved that all the terms of an oval polynomial have even degree. It is suitable for efficient computer searches.     

Collineation groups of translation planes admitting hyperbolic Buekenhout or parabolic Buekenhout-Metz unitals

It is shown that for every semifield spread in PG(3,q) and for every parabolic Buekenhout-Metz unital, there is a collineation group of the associated translation plane that acts transitively and regularly on the affine points of the parabolic unital. Conversely, any spread admitting such a group is shown to be a semifield spread. For hyperbolic Buekenhout unitals, various collineation groups of translation planes admitting such unitals and the associated planes are determined.     

A new semifield of order 2

In [G. Lunardon, Semifields and linear sets of PG(1,q^t), Quad. Mat., Dept. Math., Seconda Univ. Napoli, Caserta (in press)], G. Lunardon has exhibited a construction method yielding a theoretical family of semifields of order q²ⁿ,n>1 and n odd, with left nucleus F_qⁿ, middle and right nuclei both F_q² and center F_q. When n=3 this method gives an alternative construction of a family of semifields described in [N.L. Johnson, G. Marino, O. Polverino, R. Trombetti, On a generalization of cyclic semifields, J. Algebraic Combin. 26 (2009), 1-34], which generalizes the family of cyclic semifields obtained by Jha and Johnson in [V. Jha, N.L. Johnson, Translation planes of large dimension admitting non-solvable groups, J. Geom. 45 (1992), 87-104]. For n>3, no example of a semifield belonging to this family is known.In this paper we first prove that, when n>3, any semifield belonging to the family introduced in the second work cited above is not isotopic to any semifield of the family constructed in the former. Then we construct, with the aid of a computer, a semifield of order 2¹⁰ belonging to the family introduced by Lunardon, which turns out to be non-isotopic to any other known semifield.     

On 1-Systems of Q(6, q), q Even

In this paper, a method is developed to study locally hermitian 1-systems of Q(6, q), q even, by associating a kind of flock in PG(4, q) to them. This method is applied to a known locally hermitian 1-system of Q(6, 2^2e ), which was discovered by Offer as a spread of the hexagon H(2^2e ). The results concerning this spread appear to be suitable for generalization and enable us to find new classes of 1-systems of Q(6, q), q even. We also prove that a locally hermitian 1-system of Q(6, q), q even, which is not contained in a 5-dimensional subspace, is semi-classical if and only if it belongs to the new classes we describe. Finally, from the new classes of 1-systems arise new classes of semipartial geometries.     

On translation planes of orderq 2 that admit a group of orderq 2 (q−1); bartolone's theorem

This article is concerned with translation planesP of orderq ² and kernelK isomorphic toG F(q). IfP admits a collineation groupG in the linear translation. complement and the order ofG K/K isq ²(q?1) then it is shown thatP is either a semifield plane or is a Lüneburg-Tits, Walker or Betten plane. This generalizes earlier work of Bartolone.     

Symplectic semifield spreads of <Emphasis Type="Italic">PG</Emphasis>(5, <Emphasis Type="Italic">q</Emphasis>) and the veronese surface

In this paper we show that starting from a symplectic semifield spread S{\mathcal{S}} of PG(5, q), q odd, another symplectic semifield spread of PG(5, q) can be obtained, called the symplectic dual of S{\mathcal{S}}, and we prove that the symplectic dual of a Desarguesian spread of PG(5, q) is the symplectic semifield spread arising from a generalized twisted field. Also, we construct a new symplectic semifield spread of PG(5, q) (q = s ², s odd), we describe the associated commutative semifield and deal with the isotopy issue for this example. Finally, we determine the nuclei of the commutative pre-semifields constructed by Zha et al. (Finite Fields Appl 15(2):125–133, 2009).     

Spreads of T 2(o), α-flocks and Ovals

We establish a representation of a spread of the generalized quadrangle T ₂(0), o an oval of PG(2, q), q even, in terms of a certain family of q ovals of PG(2, q) and investigate the properties of this representation. Using this representation we show that to every flock of a translation oval cone in PG(3, q) (-flock), q even, there corresponds a spread of T ₂(o) for an oval o determined by the -flock. This gives constructions of new spreads of T ₂(o), for certain ovals o, and in some cases solves the existence problem for spreads. It also provides a geometrical characterization of the ovals associated with a flock of a quadratic cone.     

The embedding of (0, α)-geometries in PG(n, q): part II

In Part I we obtained results about the embedding of (0, α)-geometries in PG(3, q). Here we determine all (0, α)-geometries with q+1 points on a line, which are embedded in PG(n, q), n>3 and q>2. As a particular case all semi partial geometries with parameters s=q,t,α(>1),μ, which are embeddable in PG(n, q), q≠2, are obtained. We also prove some theorems about the embedding of (0, 2)-geometries in PG(n, 2): we show that without loss of generality we may restrict ourselves to reduced (0, 2)-geometries, we determine all (0, 2)-geometries in PG(4, 2), and we describe an unusual embedding of U_2,3(9) in PG(5, 2).