Desarguesian Plane PG Research Articles

A characterization of the split Cayley generalized hexagon [formula omitted] using one subhexagon of order [formula omitted

In this paper, we prove that, if a generalized hexagon Γ of order q contains a subhexagon Γ ′ (of order ( 1 , q ) ) isomorphic to the incidence graph of the Desarguesian plane PG ( 2 , q ) , and if the automorphism group of Γ stabilizing Γ ′ induces all elations in PG ( 2 , q ) , then Γ must be isomorphic to the split Cayley hexagon H ( q ) .

Linear representation of Buekenhout–Metz unitals

In the linear representation of the desarguesian plane PG(2, q 2) in PG(5, q), the classical unital (the hermitian curve) is represented by an elliptic quadric ruled by lines of a normal spread. In this paper, we prove that a Buekenhout–Metz unital U in PG(2, q 2), arising from an elliptic quadric, is represented in PG(5, q) by an algebraic hypersurface of degree four minus the complement of a line in a three-dimensional subspace. Also, such a hypersurface is reducible if and only if U is classical.

Ruled cubic surfaces in PG(4, q), Baer subplanes of PG(2, q2) and Hermitian curves

In PG(2, q 2) let ℓ ∞ denote a fixed line, then the Baer subplanes which intersect ℓ ∞ in q+1 points are called affine Baer subplanes. Call a Baer subplane of PG(2, q 2) non-affine if it intersects ℓ ∞ in a unique point. It is shown by Vincenti (Boll. Un. Mat. Ital. Suppl. 2 (1980) 31) and Bose et al. (Utilitas Math. 17 (1980) 65) that non-affine Baer subplanes of PG(2, q 2) are represented by certain ruled cubic surfaces in the André/Bruck and Bose representation of PG(2, q 2) in PG(4, q) (Math. Z. 60 (1954) 156; J. Algebra 1 (1964) 85; J. Algebra 4 (1966) 117). The André/Bruck and Bose representation of PG(2, q 2) involves a regular spread in PG(3, q). For a fixed regular spread S , it is known that not all ruled cubic surfaces in PG(4, q) correspond to non-affine Baer subplanes of PG(2, q 2) in this manner. In this paper, we prove a characterisation of ruled cubic surfaces in PG(4, q) which represent non-affine Baer subplanes of the Desarguesian plane PG(2, q 2). The characterisation relies on the ruled cubic surfaces satisfying a certain geometric condition. This result and the corollaries obtained are then applied to give a geometric proof of the result of Metsch (London Mathematical Society Lecture Note Series, Vol. 245, Cambridge University Press, Cambridge, 1997, p. 77) regarding Hermitian unitals; a result which was originally proved in a coordinate setting.

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).

Blocking Sets of Size qt+qt−1+1

We prove that in the desarguesian plane PG(2, qt) (t>4) there are at least three inequivalent blocking sets of size qt+qt−1+1. The first one has q+1 Rédei lines, the second one has exactly one Rédei line, and the third one is not of Rédei type. For GF(q) the largest subfield of GF(qt), our results disprove a conjecture quoted by A. Blokhuis (1998, in “Galois Geometry and Generalized Polygons,” Gent).

The André/Bruck and Bose representation in {${\rm PG}(2h,q)$}: unitals and Baer subplanes

Many authors have used the Andre/Bruck and Bose representation of PG(2, q2) in PG(4, q) to study objects in the Desarguesian plane with great success. This paper looks at the Andre/Bruck and Bose representation of the Desarguesian plane PG(2, qh) in PG(2h, q) in order to determine whether this higher dimensional representation provides additional information about objects in the plane. In particular, we look at the representation of unitals and Baer subplanes in this setting.

Maximal arcs in Desarguesian planes of odd order do not exist

Forq an odd prime power, and 1<n<q, the Desarguesian planePG(2,q) does not contain an(nq−q+n,n)-arc.

Nuclei of point sets of sizeq+1 contained in the union of two lines inPG(2,q)

We give a complete classification for pairs ( $$\mathcal{N}$$ (ℬ),ℬ) where $$\mathcal{N}$$ (ℬ) is the set of all nuclei of a set ℬ ofq+1 not collinear points contained in the union of two lines in a desarguesian planePG(2,q) of orderq. We also obtain some new results concerning blocking sets of Redei type and certain point-sets of type [0,1,m,n] inPG(2, q).

Blocking Sets in Desarguesian Projective Planes

Using theorems of Redéi, and of Brouwer and Schrijver, and Jamison, it is proved that a non-trivial blocking set in a desarguesian projective plane of order q has at least q + √(2q) + 1 points, if q is at least 7, odd and not a square and q ¦ 27. Further one can show that non-trivial blocking sets in the desarguesian planes PG(2, 11) and PG(2, 13) have at least 18 resp. 21 points, and this is best possible. In addition a nice description of a blocking set of size qt + qt−1 in the desarguesian plane PG(2, qt) is given, where q is some prime power.