{
  "id": "math/0204002",
  "version": "v1",
  "published": "2002-03-29T22:03:50.000Z",
  "updated": "2002-03-29T22:03:50.000Z",
  "title": "Bertini theorems over finite fields",
  "authors": [
    "Bjorn Poonen"
  ],
  "comment": "22 pages, Latex 2e",
  "categories": [
    "math.AG",
    "math.NT"
  ],
  "abstract": "Let X be a smooth quasiprojective subscheme of P^n of dimension m >= 0 over F_q. Then there exist homogeneous polynomials f over F_q for which the intersection of X and the hypersurface f=0 is smooth. In fact, the set of such f has a positive density, equal to zeta_X(m+1)^{-1}, where zeta_X(s)=Z_X(q^{-s}) is the zeta function of X. An analogue for regular quasiprojective schemes over Z is proved, assuming the abc conjecture and another conjecture.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2002-03-29T22:03:50.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14J70",
      "11M38",
      "11M41",
      "14G40",
      "14N05"
    ],
    "keywords": [
      "finite fields",
      "bertini theorems",
      "smooth quasiprojective subscheme",
      "abc conjecture",
      "regular quasiprojective schemes"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 22,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2002math......4002P"
    }
  }
}