{
  "id": "1907.11750",
  "version": "v1",
  "published": "2019-07-26T18:44:04.000Z",
  "updated": "2019-07-26T18:44:04.000Z",
  "title": "On the codimension of the singular locus",
  "authors": [
    "David Kazhdan",
    "Tamar Ziegler"
  ],
  "categories": [
    "math.AG",
    "math.CO"
  ],
  "abstract": "Let $k$ be a field and $V$ an $k$-vector space. For a family $\\bar P=\\{ P_i\\}, \\ 1\\leq i\\leq c, $ of polynomials on $V$, we denote by $\\mathbb X _{\\bar P}\\subset V$ the subscheme defined by the ideal $(\\{ P_i\\}_{1\\leq i\\leq c})$. We show the existence of $\\gamma (c,d)$ such that varieties $\\mathbb X _{\\bar P}$ are smooth outside of codimension $m$, if $deg(P_i)\\leq d$ and $r(\\bar P)\\geq \\gamma (d,c) (1+m)^{\\gamma (d,c)}$, under the condition that either $char (k)=0$ or $char (k)>\\max (d, |\\bar d|)$, where $|\\bar d| = \\sum_{i=1}^c (d_i-1)$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2019-07-26T18:44:04.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "singular locus",
      "codimension",
      "vector space",
      "smooth outside"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}