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Parallelisms of quadric sets. (English) Zbl 1305.51007

For a field \(K\), and a flock of a quadratic cone, an elliptic quadric or a hyperbolic quadric in \(PG(3,K)\), a parallelism refers to a set of mutually disjoint flocks, whose union is a complete cover of the set of conics of plane intersections of the quadric. The authors show that every flock of a hyperbolic quadric \(H\) and every flock of a quadratic cone \(C\) in \(PG(3,K)\) is in a transitive parallelism of \(H\) or \(C\) respectively. Additionally, it is shown that it is possible to have parallelisms of quadratic cones by maximal partial flocks. Parallelisms by \(\alpha\)-flokki of \(\alpha\)-cones, which are a generalization of quadratic cones, are also considered. The authors end by stating four open problems concerning parallelisms.

MSC:

51E20 Combinatorial structures in finite projective spaces
51A15 Linear incidence geometric structures with parallelism
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