{
  "id": "0712.0956",
  "version": "v1",
  "published": "2007-12-06T18:48:30.000Z",
  "updated": "2007-12-06T18:48:30.000Z",
  "title": "Affine Algebraic Varieties",
  "authors": [
    "Jing Zhang"
  ],
  "comment": "Welcome comments",
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "In this paper, we give new criteria for affineness of a variety defined over $\\Bbb{C}$. Our main result is that an irreducible algebraic variety $Y$ (may be singular) of dimension $d$ ($d\\geq 1$) defined over $\\Bbb{C}$ is an affine variety if and only if $Y$ contains no complete curves, $H^i(Y, {\\mathcal{O}}_Y)=0$ for all $i>0$ and the boundary $X-Y$ is support of a big divisor, where $X$ is a projective variety containing $Y$. We construct three examples to show that a variety is not affine if it only satisfies two conditions among these three conditions. We also give examples to demonstrate the difference between the behavior of the boundary divisor $D$ and the affineness of $Y$. If $Y$ is an affine variety, then the ring $\\Gamma (Y, {\\mathcal{O}}_Y)$ is noetherian. However, to prove that $Y$ is an affine variety, we do not start from this ring. We explain why we do not need to check the noetherian property of the ring $\\Gamma (Y, {\\mathcal{O}}_Y)$ directly but use the techniques of sheaf and cohomology.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2007-12-06T18:48:30.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J10",
      "14J30",
      "32E10"
    ],
    "keywords": [
      "affine algebraic varieties",
      "affine variety",
      "affineness",
      "noetherian property",
      "main result"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2007arXiv0712.0956Z"
    }
  }
}