Quadratic solutions of quadratic forms

The aim of this note is to study solutions of a homogeneous quadratic equation q(x(0), . . . ,x(n)) = 0, defined over a field k, where the x(i) are themselves homogeneous polynomials of some degree d in r vertical bar 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 P-1 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 P-2 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.