{
  "id": "1607.01276",
  "version": "v1",
  "published": "2016-07-05T14:44:41.000Z",
  "updated": "2016-07-05T14:44:41.000Z",
  "title": "Quadratic solutions of quadratic forms",
  "authors": [
    "János Kollár"
  ],
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "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.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2016-07-05T14:44:41.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14G05",
      "11E04",
      "14J26",
      "14J70",
      "14N15",
      "14P05"
    ],
    "keywords": [
      "quadratic forms",
      "quadratic solutions",
      "orthogonal grassmannian",
      "rational maps",
      "homogeneous quadratic equation"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}