A Survey on Varieties of PG(4, q) and Baer Subplanes of Translation Planes

Vincenti, Rita

doi:10.1016/s0304-0208(08)73355-3

Cited by 9 publications

(7 citation statements)

References 6 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…It follows that a line of PG(4, q)\Eoo corresponds to a Baer subline of PG(2, q2) with a point on £~. The converse of each of these results is true, (see [27,Prop. 1]).…”

Section: Preliminary Resultsmentioning

confidence: 86%

“…Under this representation of PG(2, q2) in PG(4, q), a plane of PG(4, q)\E~ which does not contain any line of S corresponds to a Baer subplane of PG(2, qa) with q + 1 points on g~, (see [27,Prop. 2]).…”

Section: Preliminary Resultsmentioning

confidence: 99%

“…However our current method does not extend to prove (1.3) for unitals in these non-Desarguesian planes, for we rely on the fact that the Baer sublines in PG(2, q2) with a point on the line at infinity correspond exactly to the lines of PG(4, q)\PG (3, q). This result need not hold in a non-Desarguesian translation plane; in particular results of Vincenti [27,Prop. 2] and Freeman [13,Th.…”

Section: []mentioning

confidence: 97%

See 2 more Smart Citations

Characterizations of Buekenhout-Metz unitals

1996

View full text Add to dashboard Cite

We give a characterization of the Buekenhout-Metz unitals in PG(2, q2), in the cases that q is even or q = 3, in terms of the secant lines through a single point of the unital. With the addition of extra conditions, we obtain further characterizations of Buekenhout-Metz unitals in PG(2, q2), for all q. As an application, we show that the dual of a Buekenhout-Metz unital in PG(2, q2) is a Buekenhout-Metz unital.Mathematics Subject Classification (1991): 51E20.

show abstract

“…It follows that a line of PG(4, q)\Eoo corresponds to a Baer subline of PG(2, q2) with a point on £~. The converse of each of these results is true, (see [27,Prop. 1]).…”

Section: Preliminary Resultsmentioning

confidence: 86%

Section: Preliminary Resultsmentioning

confidence: 99%

Section: []mentioning

confidence: 97%

See 1 more Smart Citation

Characterizations of Buekenhout-Metz unitals

1996

View full text Add to dashboard Cite

show abstract

“…A Baer subline M of PG(2, q 2 ) which is not contained in L ∞ corresponds either to a line of PG(4, q) \ Σ ∞ or to a nondegenerate conic (actually, an ellipse) in one of the planes of PG(4, q) \ Σ ∞ meeting Σ ∞ in a spread line, accordingly as M meets L ∞ in one or zero points. Conversely, any line of PG(4, q) \ Σ ∞ represents a Baer subline of PG(2, q 2 ) which has a point on L ∞ (see [21]). …”

Section: Introduction and Preliminary Resultsmentioning

confidence: 99%

A combinatorial characterization of classical unitals

Aguglia¹,

Ebert²

2002

Archiv der Mathematik

View full text Add to dashboard Cite

In this paper we give a characterization of classical unitals in terms of a configuration pattern formed by the feet of a unital U embedded in PG(2, q 2 ), q > 2. We show that a necessary and sufficient condition for U to be classical is the existence of two points p 0 , p 1 ∈ U with tangent lines L 0 and L 1 , respectively, such that for all points r ∈ L 0 \ { p 0 } and s ∈ L 1 \ { p 1 } the corresponding feet are collinear. Introduction and preliminary results.A unital or Hermitian arc in PG(2, q 2 ), q any prime power, is a set U of q 3 + 1 points such that every line of PG(2, q 2 ) meets U in either 1 or q + 1 points. We call a line a tangent or a secant of U accordingly as its intersection with U consists of 1 or q + 1 points. Through each point of U there pass q 2 secants and one tangent line, whereas through each point r / ∈ U there pass q + 1 tangents and q 2 − q secants. The feet of a point r / ∈ U are the q + 1 points of U that lie on the q + 1 tangents through r. For any prime power q the absolute points of a unitary polarity δ of PG(2, q 2 ) give an example of unital. Such a unital is called a classical unital or Hermitian curve.In 1976 Buekenhout [7] proved the existence of a parabolic unital in every translation plane of order q 2 with kernel containing GF(q). The term parabolic means that the line at infinity is a tangent of the unital. In [7] Buekenhout also showed that every derivable translation plane of order q 2 , with GF(q) in its kernel, contains a hyperbolic unital; that is, the line at infinity is a secant of the unital.We now briefly describe Buekenhout's two constructions, restricting ourselves to unitals embedded in PG(2, q 2 ). To do this, we make use of the representation of PG(2, q 2 ) in PG(4, q) due to André [1] and Bruck and Bose [8], [9]. Let PG(4, q) be a projective 4-space over the finite field GF(q), and let S be a regular spread of a fixed hyperplane Σ ∞ = PG(3, q) of PG(4, q). Then PG(2, q 2 ) can be represented by taking the points of PG(4, q) \ Σ ∞ as its affine points, the lines of S as its points at infinity, the planes of PG(4, q) \ Σ ∞ which meet Σ ∞ in a line of S as its extended affine lines, and S as its line L ∞ at infinity. If U is a nondegenerate quadric in PG(4, q) which meets Σ ∞ in a doubly-ruled quadric Q and if one of the reguli ruling this quadric Q is a regulus contained in the spread S , then U corresponds to a unital U in PG(2, q 2 ) and L ∞ is a secant line to U. Similarly, if U is a cone in PG(4, q) whose base

show abstract

“…Proof A plane II in PG(4, q) that meets H in a line not contained in the spread S corresponds to a Baer subplane of the translation plane 7r ( [25]). The points of the corresponding Baer subplane are given by the points of II\H, together with the points corresponding to the q + 1 spread elements that 1I meets.…”

Section: Lemma 2 Let 7r Be a Translation Plane Containing A Thas Maxmentioning

confidence: 99%

Some inherited maximal arcs in derived dual translation planes

Hamilton

1995

Geom Dedicata

View full text Add to dashboard Cite

Abstract. In 1974 J. A. Thas constructed a class of maximal arcs in certain translation planes of order q2. In this paper a new class of maximal arcs is constructed in certain derived dual translation planes that are inherited from the duals of the Thas maximal arcs. It is noted that some (but not all) of the maximal arcs are isomorphic to a class constructed by the author. (1991): 51E20. Mathematics Subject Classification

show abstract

A Survey on Varieties of PG(4, q) and Baer Subplanes of Translation Planes

Cited by 9 publications

References 6 publications

Characterizations of Buekenhout-Metz unitals

Characterizations of Buekenhout-Metz unitals

A combinatorial characterization of classical unitals

Some inherited maximal arcs in derived dual translation planes

Contact Info

Product

Resources

About