On sets of planes in projective spaces intersecting mutually in one point
Let \({\mathcal P}\) be a projective space. A set \({\mathcal K}\) of projective subplanes of \(\mathcal P\) is called a Klein set, if any two of these subplanes intersect in exactly one point. The projective subspace that is spanned by all of the intersection points of any two distinct planes of \(\mathcal K\) is called the base space of \(\mathcal K\) and its dimension is called the base dimension of \(\mathcal K\). The authors give a complete description of Klein sets with base dimension 3. Determining Klein sets of base dimension 4 is shown to be equivalent to classifying partial spreads in a four-dimensional projective space. Klein sets of base dimension 5 are classified in the case when \(\mathcal K\) is not contained in its base space. If \(\mathcal K\) is contained in its base space, the authors prove for \({\mathcal P} = \text{ PG}(5,q)\), \(q \geq 3\), and \(|\mathcal K|\geq 3(q^2+q+1)\) that \(\mathcal K\) is contained in a hyperbolic Klein set, i.e. a Klein set that arises from a hyperbolic quadric in \({\mathcal P}\). Finally, the authors classify all Klein sets with a base dimension greater than 5. It turns out that in this case there are, up to isomorphism, only three Klein sets, all of them being related to the Fano plane.
scientific article; zbMATH DE number 1384068
scientific article