The uniqueness of the 1-system of Q-(7, q), q even

For q odd, it was shown in [3] that the elliptic quadric Q−(7, q) possesses a unique 1-system, the so-called classical 1-system. Here, the same result will be obtained for even q. 1 Basic properties of 1-systems of Q−(7, q) A 1-system M of the elliptic quadric Q−(7, q) is a set {L0, L1, . . . , Lq4} of q4+1 lines of Q−(7, q) with the property that every plane of Q−(7, q) containing a line Li of M has an empty intersection with (L0 ∪ L1 ∪ . . . ∪ Lq4) \ Li. We denote the union of all elements of M by M̃. Concerning the generators of Q−(7, q), which are planes, the following result is shown by Shult and Thas in [5] in a more general context; here it is stated for 1-systems of Q−(7, q) in particular. Theorem 1.1 (Shult and Thas [5]) If M is a 1-system of the elliptic quadric Q−(7, q), then every generator of Q−(7, q) contains exactly q + 1 points of M̃. In [2] a similar result for lines of Q−(7, q) is obtained, as it is shown that every line of Q−(7, q) has 0, 1, 2 or q + 1 points in common with M̃, where the latter occurs if and only if the line belongs to M. In combination with Theorem 1.1, this implies that every totally singular plane of Q−(7, q) either contains a line of M, or a (q + 1)-arc of points of M̃. Let M be an arbitrary 1-system of Q−(7, q). If L1, L2 and L3 are arbitrary lines of M, then 〈L1, L2〉 is 3-dimensional and it intersects Q−(7, q) in a hyperbolic quadric Q+(3, q). This hyperbolic quadric Q+(3, q) contains no points ∗The first author is Research Assistant of the Fund for Scientific Research – Flanders (Bel-