Quadratic solutions of quadratic forms

TL;DR

This paper investigates solutions to quadratic equations involving polynomial variables over fields, focusing on the geometry of rational maps from projective spaces to quadrics, especially in the case of degree 2 maps to 4-dimensional quadrics.

Contribution

It provides a detailed description of the parameter spaces of degree 2 maps from projective spaces to quadrics, extending classical methods and analyzing the structure of these spaces, particularly for 4-dimensional quadrics.

Findings

The space of degree 2 maps from P^2 to quadrics has 5 irreducible components in the 4-dimensional case.

Classical geometric methods involving the Veronese surface are used to analyze these maps.

Results contribute to understanding surfaces containing multiple circles through each point in the real case.

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.