2 research outputs found
Eggs in PG(4n-1, q), q even, containing a pseudo-pointed conic
An ovoid of \PG(3,q) can be defined as a set of points with the property that
every three points span a plane and at every point there is a unique
tangent plane. In 2000 M. R. Brown (\cite{BROWN2000})
proved that if an ovoid of \PG(3,q), even,
contains a pointed conic, then either and the ovoid is an elliptic quadric, or and the ovoid is a Tits ovoid.
Generalising the definition of an ovoid to a set of -spaces of
\PG(4n-1,q) J. A. Thas \cite{THAS1971} introduced the notion of pseudo-ovoids
or eggs: a set of -spaces in \PG(4n-1,q), with the property that
any three egg elements span a -space and at every egg element there is
a unique tangent -space.
We prove that an egg in \PG(4n-1,q), even, contains a pseudo pointed conic, that is,
a pseudo-oval arising from a pointed conic of \PG(2,q^n), even,
if and only if the egg is elementary and the ovoid is either an elliptic quadric in \PG(3,4) or a Tits ovoid in \PG(3,8)