On mm-ovoids of Q+(7,q)Q^+(7,q) with qq odd

In this paper, we provide a construction of $(q+1)$-ovoids of the hyperbolic quadric $Q^+(7,q)$, $q$ an odd prime power, by glueing $(q+1)/2$-ovoids of the elliptic quadric $Q^-(5,q)$. This is possible by controlling some intersection properties of (putative) $m$-ovoids of elliptic quadrics. It yields eventually $(q+1)$-ovoids of $Q^+(7,q)$ not coming from a $1$-system. Secondly, we also construct $m$-ovoids for $m \in \{ 2,4,6,8,10\}$ in $Q^+(7,3)$. Therefore we first investigate how to construct spreads of \pg(3,q) that have as many secants to an elliptic quadric as possible

Locally hermitian 1-systems of Q(+)(7,q)

AbstractA flock of a cone in PG(5,q) with a line as vertex and a hyperbolic quadric Q+(3,q) as base is associated with every locally hermitian 1-system of Q+(7,q) and conversely, so that the two objects are equivalent. We construct an example of such a flock, starting from a Segre variety S1;2, and study the corresponding 1-system of Q+(7,q). Locally hermitian semiclassical 1-systems of Q+(7,q), which are not contained in a hyperplane of PG(7,q), are characterized in terms of their flock. Finally, the previously known locally hermitian semiclassical 1-systems of Q+(7,q) are investigated and it seems that many new examples can be found