A characterization of quadrics by intersection numbers

This work is inspired by a paper of Hertel and Pott on maximum non-linear functions (Hertel and Pott, A characterization of a class of maximum non-linear functions. Preprint, 2006). Geometrically, these functions correspond with quasi-quadrics; objects introduced in De Clerck et al. (Australas J Combin 22:151–166, 2000). Hertel and Pott obtain a characterization of some binary quasi-quadrics in affine spaces by their intersection numbers with hyperplanes and spaces of codimension 2. We obtain a similar characterization for quadrics in projective spaces by intersection numbers with low-dimensional spaces. Ferri and Tallini (Rend Mat Appl 11(1): 15–21, 1991) characterized the non-singular quadric Q(4,q) by its intersection numbers with planes and solids. We prove a corollary of this theorem for Q(4,q) and then extend this corollary to all quadrics in PG(n,q),n ≥ 4. The only exceptions occur for q even, where we can have an oval or an ovoid as intersection with our point set in the non-singular part.      

Transitive Arcs in Planes of Even Order

When one considers the hyperovals inPG(2,q),qeven,q>2, then the hyperoval inPG(2, 4) and the Lunelli-Sce hyperoval inPG(2, 16) are the only hyperovals stabilized by a transitive projective group [10]. In both cases, this group is an irreducible group fixing no triangle in the plane of the hyperoval, nor in a cubic extension of that plane. Using Hartley's classification of subgroups ofPGL₃(q),qeven [6], allk-arcs inPG(2,q) fixed by a transitive irreducible group, fixing no triangle inPG(2,q) or inPG(2,q³), are determined. This leads to new 18-, 36- and 72-arcs inPG(2,q),q=2^2h. The projective equivalences among the arcs are investigated and each section ends with a detailed study of the collineation groups of these arcs.     

A characterization of the family of external lines to a quadratic cone of PG(3, q), q

In this paper we characterize the family of external lines to a quadratic cone of PG(3, q), q odd, by their intersection properties with points and planes of the space.     

Some Characterizations of Finite Hermitian Veroneseans

We characterize the finite Veronesean of all Hermitian varieties of PG(n,q²) as the unique representation of PG(n,q²) in PG(d,q), d n(n+2), where points and lines of PG(n,q²) are represented by points and ovoids of solids, respectively, of PG(d,q), with the only condition that the point set of PG(d,q) corresponding to the point set of PG(n,q²) generates PG(d,q). Using this result for n=2, we show that is characterized by the following properties: (1) ; (2) each hyperplane of PG(8,q) meets in q²+1, q³+1 or q³+q²+1 points; (3) each solid of PG(8,q) having at least q+3 points in common with shares exactly q²+1 points with it.51E24     

On the Geometry of Hermitian Matrices of Order Three Over Finite Fields

Some geometry of Hermitian matrices of order three over GF(q²) is studied. The variety coming from rank 2 matrices is a cubic hypersurface M₇³of PG(8,q ) whose singular points form a variety H corresponding to all rank 1 Hermitian matrices. BesideM₇³ turns out to be the secant variety of H. We also define the Hermitian embedding of the point-set of PG(2, q²) whose image is exactly the variety H. It is a cap and it is proved that PGL(3, q²) is a subgroup of all linear automorphisms of H. Further, the Hermitian lifting of a collineation of PG(2, q²) is defined. By looking at the point orbits of such lifting of a Singer cycle of PG(2, q²) new mixed partitions of PG(8,q ) into caps and linear subspaces are given.     

Complete Spans on Hermitian Varieties

Let L be a general linear complex in PG(3, q) for any prime power q. We show that when GF(q) is extended to GF(q ²), the extended lines of L cover a non-singular Hermitian surface H ? H(3, q ²) of PG(3, q ²). We prove that if Sis any symplectic spread PG(3, q), then the extended lines of this spread form a complete (q ² + 1)-span of H. Several other examples of complete spans of H for small values of q are also discussed. Finally, we discuss extensions to higher dimensions, showing in particular that a similar construction produces complete (q ³ + 1)-spans of the Hermitian variety H(5, q ²).     

On the intersection of two subgeometries of PG(n, q)

The study of the intersection of two Baer subgeometries of PG(n, q), q a square, started in Bose et al. (Utilitas Math 17, 65–77, 1980); Bruen (Arch Math 39(3), 285–288, (1982). Later, in Svéd (Baer subspaces in the n-dimensional projective space. Springer-Verlag) and Jagos et al. (Acta Sci Math 69(1–2), 419–429, 2003), the structure of the intersection of two Baer subgeometries of PG(n, q) has been completely determined. In this paper, generalizing the previous results, we determine all possible intersection configurations of any two subgeometries of PG(n, q).      

A characterization of the Grassmann embedding of H(q), with q even

In this note, we characterize the Grassmann embedding of H(q), q even, as the unique full embedding of H(q) in PG(12, q) for which each ideal line of H(q) is contained in a plane. In particular, we show that no such embedding exists for H(q), with q odd. As a corollary, we can classify all full polarized embeddings of H(q) in PG(12, q) with the property that the lines through any point are contained in a solid; they necessarily are Grassmann embeddings of H(q), with q even.     

Blocking sets and semifields

We find a relationship between semifield spreads of PG(3,q), small Rédei minimal blocking sets of PG(2,q²), disjoint from a Baer subline of a Rédei line, and translation ovoids of the hermitian surface H(3,q²).     

A Combinatorial Characterization of Hermitian Curves

A unital U with parameter q is a 2 – (q ³ + 1, q + 1, 1) design. If a point set U in PG(2, q ²) together with its (q + 1)-secants forms a unital, then U is called a Hermitian arc. Through each point p of a Hermitian arc H there is exactly one line L having with H only the point p in common; this line L is called the tangent of H at p. For any prime power q, the absolute points and nonabsolute lines of a unitary polarity of PG(2, q ²) form a unital that is called the classical unital. The points of a classical unital are the points of a Hermitian curve in PG(2, q ²).Let H be a Hermitian arc in the projective plane PG(2, q ²). If tangents of H at collinear points of H are concurrent, then H is a Hermitian curve. This result proves a well known conjecture on Hermitian arcs.     

The classification of spreads in PG(3, q) admitting linear groups of order q(q+1), I. odd order

A classification is given of all spreads in PG(3, q), q = p^r, p odd, whose associated translation planes admit linear collineation groups of order q(q +1) such that a Sylow p-subgroup fixes a line and acts non-trivially on it.The authors are indebted to T. Penttila for pointing out the special examples of conical flock translation planes of order q² that admit groups of order q(q+1), when q = 23 or 47.     

A nearfield-free definition of regular Hughes planes and some embeddings of PG(3,q) in Hughes planes of order q 4

We give a nearfield-free definition of some finite and infinite incidence systems by means of half-points and half-lines and show that they are projective planes. We determine a planar ternary ring for these planes and use it to determine the full collineation group and to demonstrate some embeddings of these planes among themselves. We show that these planes include all finite regular Hughes planes and many infinite ones. We also show that PG(3, q) embeds in Hu(q ⁴) (and show infinite versions of this embedding). Dan Hughes 80th Birthday.     

Characterizing geometric designs, II

We provide a characterization of the classical point-line designs PG₁(n,q), where n?3, among all non-symmetric 2-(v,k,1)-designs as those with the maximal number of hyperplanes. As an application of this result, we characterize the classical quasi-symmetric designs PGn−2(n,q), where n?4, among all (not necessarily quasi-symmetric) designs with the same parameters as those having line size q+1 and all intersection numbers at least qn−4+?+q+1. Finally, we also give an explicit lower bound for the number of non-isomorphic designs having the same parameters as PG₁(n,q); in particular, we obtain a new proof for the known fact that this number grows exponentially for any fixed value of q.     

Generalised dual arcs and Veronesean surfaces, with applications to cryptography

We start by defining generalised dual arcs, the motivation for defining them comes from cryptography, since they can serve as a tool to construct authentication codes and secret sharing schemes. We extend the characterisation of the tangent planes of the Veronesean surface in PG(5,q), q odd, described in [J.W.P. Hirschfeld, J.A. Thas, General Galois Geometries, Oxford Math. Monogr., Clarendon Press/Oxford Univ. Press, New York, 1991], as a set of q²+q+1 planes in PG(5,q), such that every two intersect in a point and every three are skew. We show that a set of q²+q planes generating PG(5,q), q odd, and satisfying the above properties can be extended to a set of q²+q+1 planes still satisfying all conditions. This result is a natural generalisation of the fact that a q-arc in PG(2,q), q odd, can always be extended to a (q+1)-arc. This extension result is then used to study a regular generalised dual arc with parameters (9,5,2,0) in PG(9,q), q odd, where we obtain an algebraic characterisation of such an object as being the image of a cubic Veronesean.     

Unitals in Finite Desarguesian Planes

We show that a suitable 2-dimensional linear system of Hermitian curves of PG(2,q ²) defines a model for the Desarguesian plane PG(2,q). Using this model we give the following group-theoretic characterization of the classical unitals. A unital in PG(2,q ²) is classical if and only if it is fixed by a linear collineation group of order 6(q + 1)² that fixes no point or line in PG(2,q ²).     

Some characterizations of quasi-symmetric designs with a spread

The designPG ₂ (4,q) of the points and planes ofPG (4,q) forms a quasi-symmetric 2-design with block intersection numbersx=1 andy=q+1. We give some characterizations of quasi-symmetric designs withx=1 which have a spread through a fixed point. For instance, it is proved that if such a designD is also smooth, thenDPG ₂ (4,q).     

Some two-character sets

Some new examples of two-character sets with respect to planes of PG(3, q ²), q odd, are constructed. They arise from three-dimensional hyperbolic quadrics, from the geometry of an orthogonal polarity commuting with a unitary polarity. The last examples arise from the geometry of the unitary group PSU(3, 3) acting on the split Cayley hexagon H(2).     

Blocking Sets and Derivable Partial Spreads

We prove that a GF(q)-linear Rédei blocking set of size q ^t + q ^t–1 + ··· + q + 1 of PG(2,q ^t) defines a derivable partial spread of PG(2t – 1, q). Using such a relationship, we are able to prove that there are at least two inequivalent Rédei minimal blocking sets of size q ^t + q ^t–1 + ··· + q + 1 in PG(2,q ^t), if t 4.     

Designs with block-size 6 in projective planes of characteristic 2

We construct simple 3-designs and 4-designs of block-size 6 in the classical projective planesPG(2,q),q a power of 2. All of our designs are invariant under the projective groupPGL(3,q). Aside from several infinite series of 3-designs we get some relatively small designs of independent interest, e.g. designs with parameters 4-(21, 6, 16) and 4-(73, 6, 330) defined in the planes of orders 4 and 8, respectively.     

Derivation techniques on the Hermitian surface

We discuss derivation‐like techniques for transforming one locally Hermitian partial ovoid of the Hermitian surface H(3,q²) into another one. These techniques correspond to replacing a regulus by its opposite in some naturally associated projective 3‐space PG(3,q) over a square root subfield. © 2006 Wiley Periodicals, Inc. J Combin Designs 15: 478–486, 2007