{
  "id": "2302.11470",
  "version": "v1",
  "published": "2023-02-22T16:16:33.000Z",
  "updated": "2023-02-22T16:16:33.000Z",
  "title": "Surjective morphisms from affine space to its Zariski open subsets",
  "authors": [
    "Viktor Balch Barth"
  ],
  "comment": "7 pages. Comments welcome!",
  "categories": [
    "math.AG"
  ],
  "abstract": "We prove constructively the existence of surjective morphisms from affine space onto certain open subvarieties of affine space of the same dimension. For any algebraic set $Z\\subset \\mathbb{A}^{n-2}\\subset \\mathbb{A}^{n}$, we construct an endomorphism of $\\mathbb{A}^{n}$ with $\\mathbb{A}^{n} \\setminus Z$ as its image. By Noether's normalization lemma, these results extend to give surjective maps from any $n$-dimensional affine variety $X$ to $\\mathbb{A}^{n} \\setminus Z$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2023-02-22T16:16:33.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14R10",
      "14A10"
    ],
    "keywords": [
      "affine space",
      "zariski open subsets",
      "surjective morphisms",
      "noethers normalization lemma",
      "dimensional affine variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 7,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}