Partial Flocks of Non-Singular Quadrics in PG(2r + 1, q)

We generalise the definition and many properties of partial flocks of non-singular quadrics in PG(3, q) to partial flocks of non-singular quadrics in PG(2r + 1, q).     

Partial Flocks of Quadratic Cones with a Point Vertex in Pg(n,q), n Odd

We generalise the definition and many properties of flocks ofquadratic cones in PG(3,q) to partial flocks of quadratic coneswith vertex a point in PG(p,q), for n 3 odd.     

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.     

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.     

Some p‐ranks related to a conic in PG(2, q)

Let A be the incidence matrix of lines and points of the classical projective plane PG(2, q) with q odd. With respect to a conic in PG(2, q), the matrix A is partitioned into 9 submatrices. The rank of each of these submatrices over F_q, the defining field of PG(2, q), is determined. © 2010 Wiley Periodicals, Inc. J Combin Designs 18: 224–236, 2010     

Linear Point Sets and Rédei Type k-blocking Sets in PG(n, q)

In this paper, k-blocking sets in PG(n, q), being of Rédei type, are investigated. A standard method to construct Rédei type k-blocking sets in PG(n, q) is to construct a cone having as base a Rédei type k-blocking set in a subspace of PG(n, q). But also other Rédei type k-blocking sets in PG(n, q), which are not cones, exist. We give in this article a condition on the parameters of a Rédei type k-blocking set of PG(n, q = p ^h ), p a prime power, which guarantees that the Rédei type k-blocking set is a cone. This condition is sharp. We also show that small Rédei type k-blocking sets are linear.     

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.     

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.     

Cyclic Arcs in PG(2, q)

B.C. Kestenband [9], J.C. Fisher, J.W.P. Hirschfeld, and J.A. Thas [3], E. Boros, and T. Szönyi [1] constructed complete (q ² ? q + l)-arcs in PG(2, q ²), q ≥ 3. One of the interesting properties of these arcs is the fact that they are fixed by a cyclic protective group of order q ² ? q + 1. We investigate the following problem: What are the complete k-arcs in PG(2, q) which are fixed by a cyclic projective group of order k? This article shows that there are essentially three types of those arcs, one of which is the conic in PG(2, q), q odd. For the other two types, concrete examples are given which shows that these types also occur.     

Characterizations of Hermitian varieties by intersection numbers

In this paper, we give characterizations of the classical generalized quadrangles H(3, q ²) and H(4, q ²), embedded in PG(3, q ²) and PG(4, q ²), respectively. The intersection numbers with lines and planes characterize H(3, q ²), and H(4, q ²) is characterized by its intersection numbers with planes and solids. This result is then extended to characterize all Hermitian varieties in dimension at least 4 by their intersection numbers with planes and solids.      

On the Twisted Cubic of PG(3, q)

In this paper we classify the lines of PG(3, q) whose points belong to imaginary chords of the twisted cubic of PG(3, q). Relying on this classification result, we obtain a complete classification of semiclassical spreads of the generalized hexagon H(q).     

Flocks and ovals

An infinite family of q-clans, called the Subiaco q-clans, is constructed for q=2^e. Associated with these q-clans are flocks of quadratic cones, elation generalized quadrangles of order (q ², q), ovals of PG(2, q) and translation planes of order q ² with kernel GF(q). It is also shown that a q-clan, for q=2^e, is equivalent to a certain configuration of q+1 ovals of PG(2, q), called a herd.W. Cherowitzo gratefully acknowledges the support of the Australian Research Council and has the deepest gratitude and warmest regards for the Combinatorial Computing Research Group at the University of Western Australia for their congenial hospitality and moral support. I. Pinneri gratefully acknowledges the support of a University of Western Australia Research Scholarship.     

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).      

Blocking Sets Of External Lines To A Conic In PG(2,q), q ODD

We determine all point-sets of minimum size in PG(2,q), q odd that meet every external line to a conic in PG(2,q). The proof uses a result on the linear system of polynomials vanishing at every internal point to the conic and a corollary to the classification theorem of all subgroups of PGL(2,q). * Research supported by the Italian Ministry MURST, Strutture geometriche, combinatoria e loro applicazioni and by the Hungarian-Italian Intergovernemental project “Algebraic and Geometric Structures”.     

Maximal partial spreads in PG (3, q)

We transfer the whole geometry of PG(3, q) over a non-singular quadric Q_4,q of PG(4, q) mapping suitably PG(3, q) over Q_4,q. More precisely the points of PG(3, q) are the lines of Q_4,q; the lines of PG(3, q) are the tangent cones of Q_4,q and the reguli of the hyperbolic quadrics hyperplane section of Q_4,q. A plane of PG(3, q) is the set of lines of Q_4,q meeting a fixed line of Q_4,q. We remark that this representation is valid also for a projective space over any field K and we apply the above representation to construct maximal partial spreads in PG(3, q). For q even we get new cardinalities for For q odd the cardinalities are partially known.     

Maximal Partial Spreads and Flocks

Denote by a flock of a quadratic cone of PG(3,q) by S() the spread of PG(3,q) associated with and by l the common line of the base reguli. Suppose that there are two lines not transversal to a base regulus which share the same lines of S() Then we prove that is either linear or a Kantor-Knuth semifield flock. Using this property we can extend the result of J3 on derivable flocks proving that, if a set of q + 1 lines of S() defines a derivable net different from a base regulus-net, then is either linear or a Kantor-Knuth semifield flock. Moreover if l is not a component of the derivable net, then is linear.     

On 1-Blocking Sets in PG(n,q), n ≥ 3

In this article we study minimal1-blocking sets in finite projective spaces _PG(n,q),n 3. We prove that in PG(n,q ²),q = p ^h , p prime, p > 3,h 1, the second smallest minimal 1-blockingsets are the second smallest minimal blocking sets, w.r.t.lines, in a plane of PG(n,q ²). We also study minimal1-blocking sets in PG(n,q ³), n 3, q = p ^h, p prime, p > 3,q 5, and prove that the minimal 1-blockingsets of cardinality at most q 3 + q ² + q + ¹ are eithera minimal blocking set in a plane or a subgeometry PG(3,q).     

<Emphasis Type="Italic">mth</Emphasis>-Root subgeometry partitions

New subgeometry partitions of PG(n − 1, q ^m ) by subgeometries isomorphic to PG(n − 1, q) are constructed.      

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.     

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