On sets of \(n\)-dimensional subspaces of projective spaces intersecting mutually in an \((n-2)\)-dimensional subspace (Q1849883)

Let \({\mathcal P}\) be a projective space and let \(-1\leq k \leq n\). A set \({\mathcal E}\) of \(n\)-dimensional subspaces of \({\mathcal P}\) such that any two elements of \({\mathcal E}\) intersect precisely in some \(k\)-dimensional subspace is called an \((n,k)\)-SCID (Subspaces with Constant Intersection Dimension). Such a set \({\mathcal E}\) is called primitive, if \({\mathcal E}\) spans \({\mathcal P}\), no point of \({\mathcal P}\) is contained in all elements of \({\mathcal E}\), each element of \({\mathcal E}\) is spanned by all the intersection points contained in it, and \(\dim {\mathcal P}\geq 2n+1\). For primitive \((2,0)\)-SCIDs of any \(d\)-dimensional projective space \({\mathcal P}\) \textit{A. Beutelspacher, J. Müller} and the author of the manuscript under review proved that \(5 \leq d \leq 6\). If \(d=6\), then \(5 \leq |{\mathcal E}|\leq 6\) and there are only three nonisomorphic examples of such a set \({\mathcal E}\). If \({\mathcal P} = \)PG\((5,q)\) and \(|{\mathcal E}|\geq 3(q^2+q+1)\), then \({\mathcal E}\) is contained in a hyperbolic SCID. For more details see Geom. Dedicata 78, No. 2, 143-159 (1999; Zbl 0945.51001). Here, the remaining cases for \((n,n-2)\)-SCIDs are investigated. It turns out that there is only one primitive \((3,1)\)-SCID and there are no primitive \((n,n-2)\)-SCIDs for \(n\geq 4\).