Quadratic solutions of quadratic forms

Abstract

We study solutions of a homogeneous quadratic equation $q(x_0,\dots, x_n)=0$, defined over a field $K$, where the $x_i$ 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 $P^1$ 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 of $P^2$ to quadrics, given by degree 2 polynomials. 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.