Flags in affine planes and maximal spreads in a non-singular quadric of \(PG(4,q)\) (Q1321701)

A line spread in a non-singular quadric \(Q_{4,q}\) of the projective space \(\text{PG}(4,q)\) is a set of mutually skew lines in the quadric. A line spread which is a covering of the quadric is called total, and one which is not properly contained in another line spread is called maximal. The size of a line spread \(F\), denoted by \(| F|\) is the number of lines in the spread. There are numerous results on the size of a maximal spread and many constructions for families of particular sizes, but the full spectrum has not been determined. In this paper, a new construction is given for a family of non-total maximal spreads with \(| F|= 3q+1\), \(q\geq 4\). The construction utilizes a representation of the points of a hyperbolic quadric in the affine space \(\text{AG}(4,K)\), \(K\) any commutative field, by flags \((P,r)\) of \(\text{AG}(2,K)\), where \(r\) is a line not through the origin of the affine plane. The lines of the quadric can then be identified with certain families of flags in the affine plane. Starting with a non-degenerate conic in the plane, the representation by flags is used to construct the desired line spread. A block set for the planes of \(\text{PG}(3,q)\) can also be obtained from the centers of the pencils of lines in the constructed line spread \(F\).