The uniqueness of the 1-system of \(Q^-(7,q)\), \(q\) odd (Q1601435)

A 1-system of the elliptic quadric \(Q^-(7,q)\) is a set of \(q^4\) lines such that every plane of \(Q^-(7,q)\) containing one of the lines has empty intersection with the union of the others. The main result of the article concerns the uniqueness of such an object in \(Q^-(7,q)\) if \(q\) is odd, up to a projectivity. The authors start from a general property of \(m\)-systems in elliptic polar spaces, due to \textit{E. E. Shult} and \textit{J. A. Thas} [J. Comb. Theory, Ser. A 68, 184-204 (1994; Zbl 0824.51004)] and use elements of egg theory and some results on quadrics in \(Q^-(7,q)\) (hyperbolic quadrics \(Q^+(3,q)\) and \(Q^+(5,q)\), elliptic quadrics \(Q^-(5,q)\)).