Pseudo-embeddings and quadratic sets of quadrics
This paper gives a local characterisation of the intersection of two quadrics. More precisely, the authors consider quadratic sets of a non-singular quadric \(Q\), meaning a set of points of \(Q\) with the property that it intersects each subspace of \(Q\) in a possibly singular quadric. The standard example is then given by intersecting \(Q\) with another quadric in the same projective space. In the current paper, the quadrics that are considered are the Klein quadric \(Q^+(5,2)\), the parabolic quadric \(Q(6,2)\) and the elliptic quadric \(Q^-(7,2)\), and the result is that the quadratic sets are all given by intersecting with a quadric. It is also mentioned that this result is no longer true for \(Q^+(5,q)\) with \(q\geq 3\), showing that these small cases behave differently. The paper arose as a by-product of other investigations, related to the study of certain line sets of \(\mathsf{PG}(3,q)\), which gives points on the Klein quadric \(Q^+(5,q)\), yielding a new example. The method uses pseudo-embeddings of the geometry formed by the points of the quadric \(Q\) and its planes.
scientific article; zbMATH DE number 7459700
scientific article