TITLE:
Subplanes of PG( 2, q 3 ) and the Ruled Varieties V 2 5 of PG( 6,q )
AUTHORS:
Rita Vincenti
KEYWORDS:
Finite Geometry, Translation Planes, Spreads, Varieties
JOURNAL NAME:
Open Journal of Discrete Mathematics,
Vol.14 No.2,
April
25,
2024
ABSTRACT: In this note we study subplanes of order q of the projective plane
Π=PG(
2,
q
3
)
and the ruled varieties
V
2
5
of
Σ=PG(
6,q
)
using the spatial representation of Π in Σ, by fixing a hyperplane
Σ
′
with a regular spread of planes. First are shown some configurations of the affine q-subplanes. Then to prove that a variety
V
2
5
of Σ represents a non-affine subplane of order q of Π, after having shown basic incidence properties of it, such a variety
V
2
5
is constructed by choosing appropriately the two directrix curves in two complementary subspaces of Σ. The result can be translated into further incidence properties of the affine points of
V
2
5
. Then a maximal bundle of varieties
V
2
5
having in common one directrix cubic curve is constructed.