An ovoid of \$\PG(3,q)\$ can be defined as a set of \$q^2+1\$ 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)\$, \$q\$ even, contains a pointed conic, then either \$q=4\$ and the ovoid is an elliptic quadric, or \$q=8\$ and the ovoid is a Tits ovoid. Generalising the definition of an ovoid to a set of \$(n-1)\$-spaces of \$\PG(4n-1,q)\$ J. A. Thas \cite{THAS1971} introduced the notion of pseudo-ovoids or eggs: a set of \$q^{2n}+1\$ \$(n-1)\$-spaces in \$\PG(4n-1,q)\$, with the property that any three egg elements span a \$(3n-1)\$-space and at every egg element there is a unique tangent \$(3n-1)\$-space. We prove that an egg in \$\PG(4n-1,q)\$, \$q\$ even, contains a pseudo pointed conic, that is, a pseudo-oval arising from a pointed conic of \$\PG(2,q^n)\$, \$q\$ 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)\$.

### Eggs in PG(4n-1, q), q even, containing a pseudo-pointed conic

#### Abstract

An ovoid of \$\PG(3,q)\$ can be defined as a set of \$q^2+1\$ 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)\$, \$q\$ even, contains a pointed conic, then either \$q=4\$ and the ovoid is an elliptic quadric, or \$q=8\$ and the ovoid is a Tits ovoid. Generalising the definition of an ovoid to a set of \$(n-1)\$-spaces of \$\PG(4n-1,q)\$ J. A. Thas \cite{THAS1971} introduced the notion of pseudo-ovoids or eggs: a set of \$q^{2n}+1\$ \$(n-1)\$-spaces in \$\PG(4n-1,q)\$, with the property that any three egg elements span a \$(3n-1)\$-space and at every egg element there is a unique tangent \$(3n-1)\$-space. We prove that an egg in \$\PG(4n-1,q)\$, \$q\$ even, contains a pseudo pointed conic, that is, a pseudo-oval arising from a pointed conic of \$\PG(2,q^n)\$, \$q\$ 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)\$.
##### Scheda breve Scheda completa Scheda completa (DC)
2005
File in questo prodotto:
Non ci sono file associati a questo prodotto.
##### Pubblicazioni consigliate

I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.

Utilizza questo identificativo per citare o creare un link a questo documento: `https://hdl.handle.net/11577/156921`
• ND
• 5
• 3