Parallelisms of quadric sets

Abstract

In this article, it is shown that every flock of a hyperbolic quadric $H$ and every flock of a quadratic cone $C$ in $PG\left(3,K\right)$, for $K$ a field, is in a transitive parallelism of $H$ or $C$, respectively. Furthermore, it is shown it is possible to have parallelisms of quadratic cones by maximal partial flocks. The theory of parallelisms of quadratic cones is generalized to analogous results for parallelisms of $\alpha$-cones.