{
  "id": "1503.02173",
  "version": "v1",
  "published": "2015-03-07T14:04:33.000Z",
  "updated": "2015-03-07T14:04:33.000Z",
  "title": "Algebraic curves, rich points, and doubly-ruled surfaces",
  "authors": [
    "Larry Guth",
    "Joshua Zahl"
  ],
  "comment": "33 pages",
  "categories": [
    "math.AG",
    "cs.CG",
    "math.CO"
  ],
  "abstract": "We study the structure of collections of algebraic curves in three dimensions that have many curve-curve incidences. In particular, let $k$ be a field and let $\\mathcal{L}$ be a collection of $n$ space curves in $k^3$, with $n<\\!\\!<(\\operatorname{char}(k))^2$ or $\\operatorname{char}(k)=0$. Then either A) there are at most $O(n^{3/2})$ points in $k^3$ hit by at least two curves, or B) at least $\\Omega(n^{1/2})$ curves from $\\mathcal{L}$ must lie on a bounded-degree surface, and many of the curves must form two \"rulings\" of this surface. We also develop several new tools including a generalization of the classical flecnode polynomial of Salmon and new algebraic techniques for dealing with this generalized flecnode polynomial.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-03-07T14:04:33.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "algebraic curves",
      "rich points",
      "doubly-ruled surfaces",
      "curve-curve incidences",
      "classical flecnode polynomial"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150302173G"
    }
  }
}