The hyperbolic lines of finite symplectic spaces (Q1101956)

Let (\({\mathcal P},{\mathcal L})\) be a finite, connected partial linear space in which each pair of intersecting lines lies in a subspace which is a projective plane minus a point and all lines through it. Assume that (\({\mathcal P},{\mathcal L})\) contains at least two such planes. The author proves that either (1) (\({\mathcal P},{\mathcal L})\) is a symplectic space, i.e. for some prime power q and some integer d at least 3, (\({\mathcal P},{\mathcal L})\) is isomorphic to the partial linear space of hyperbolic lines of a symplectic geometry of the Desarguesian projective space PG(d,q); or (2) all lines of \({\mathcal L}\) contain exactly three points and so (\({\mathcal P},{\mathcal L})\) is one of the spaces discussed in a previous article by the author. The spaces in (2) are not necessarily symplectic, but all can be embedded in symplectic spaces. The proof involves the consideration of a collineation group of (\({\mathcal P},{\mathcal L})\) generated by elations.