2016/07/05 by Kollár, János · 1 citation
#11E04 #14G05 #14J26 #14J70 #14N15 #14P05 #Algebraic Geometry (math.AG) #FOS: Mathematics #Number Theory (math.NT)
paper · doi:10.48550/arxiv.1607.01276
We study solutions of a homogeneous quadratic equation q(x0,…, xn)=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 Pr 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 of P2 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.