Quadratic solutions of quadratic forms

The aim of this note is to study solutions of a homogeneous quadratic equation q(x₀,…, x_n) = 0, defined over a field k, where the xi are themselves homogeneous polynomials of some degree d in r + 1 variables. Equivalently, we are looking at rational maps from projective r-space P^r to a quadric hypersurface Q, defined over a field k. The space of maps of P¹ to a quadric Q is stably birational to Q if d is even and to the orthogonal Grassmannian of lines in Q if d is odd. Most of the paper is devoted to obtaining similar descriptions for the spaces parametrizing maps, given by degree 2 polynomials, from P² to quadrics. The most interesting case is 4-dimensional quadrics when there are 5 irreducible components. The methods are mostly classical, involving the Veronese surface, its equations and projections. In the real case, these results provide some of the last steps of a project, started by Kummer and Darboux, to describe all surfaces that contain at least 2 circles through every point.