Ovoidal fibrations in $PG(3,q),q$ even

Abstract

We prove that, given a partition of the point-set of $PG(3,q),q=2^n>2$, by ovoids $\{{\theta }_i{\}}_{i=0}^q$ of $PG(3,q)$ and a line $\ell$ of $PG(3,q)$, not tangent to ${\theta }_0$ if ${\ell }^{\perp }$ denotes the polar of $\ell$ relative to the symplectic form on $PG(3,q)$ whose isotropic lines are the tangent lines to ${\theta }_0$, then $\ell$ and ${\ell }^{\perp }$ are tangent to distinct ovoids ${\theta }_j,{\theta }_k$, both distinct from ${\theta }_0$. This uses the fact that the radical of the linear code generated by the dual duals $\ell \cup {\ell }^{\perp }$ of the hyperbolic quadrics , with $\ell$ and ${\ell }^{\perp }$ as above, is of codimension $1$

10 pages