On quadrics through five real points

Let Q1,…, Q5 be five fixed points (no four coplanar) of the real projective space S3: let s be a variable quadric surface through these points. The set of all such quadrics can be represented by the points of a real S4, in which there is a quartic primal that represents cones. The geometry of this threefold is well known in the complex case, but has hardly been considered at all in the real case: and one object of the present paper is to describe the real threefold and determine its homology groups.