{
  "id": "1411.5011",
  "version": "v1",
  "published": "2014-11-18T20:54:45.000Z",
  "updated": "2014-11-18T20:54:45.000Z",
  "title": "Degree of $\\mathbb{K}$-uniruledness of the non-properness set of a polynomial map",
  "authors": [
    "Zbigniew Jelonek",
    "Michał Lasoń"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X\\subset\\mathbb{C}^n$ be an affine variety covered by polynomial curves, and let $f:X\\rightarrow\\mathbb{C}^m$ be a generically finite polynomial map. The first author showed that the set $S_f$, of points at which $f$ is not proper, is covered by polynomial curves. We generalize this result to the field of real numbers. We prove that if $X\\subset\\mathbb{R}^n$ is a closed semialgebraic set covered by polynomial curves, and $f: X\\rightarrow\\mathbb{R}^m$ is a generically finite polynomial map, then the set $S_f$ is also covered by polynomial curves. Moreover, if $X$ is covered by curves of degree at most $d_1$, and the map $f$ has degree $d_2$, then the set $S_f$ is covered by polynomial curves of degree at most $2d_1d_2$. For the field of complex numbers we get a better bound by $d_1d_2$, which is best possible. Additionally, if $X=\\mathbb{C}^n$ or $X=\\mathbb{R}^n$, and a generically finite polynomial map $f$ has degree $d$, then the set $S_f$ is covered by polynomial curves of degree at most $d-1$. This bound is best possible.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-11-18T20:54:45.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14R25",
      "14P10",
      "14R99"
    ],
    "keywords": [
      "polynomial curves",
      "generically finite polynomial map",
      "non-properness set",
      "uniruledness",
      "complex numbers"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1411.5011J"
    }
  }
}