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

A 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.