{
  "id": "1412.5007",
  "version": "v1",
  "published": "2014-12-16T14:15:47.000Z",
  "updated": "2014-12-16T14:15:47.000Z",
  "title": "Invariants of plane curve singularities and Plücker formulas in positive characteristic",
  "authors": [
    "Hong Duc Nguyen"
  ],
  "comment": "15 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "We study classical invariants for plane curve singularities $f\\in K[[x,y]]$, $K$ an algebraically closed field of characteristic $p\\geq 0$: Milnor number, delta invariant, kappa invariant and multiplicity. It is known, in characteristic zero, that $\\mu(f)=2\\delta(f)-r(f)+1$ and that $\\kappa(f)=2\\delta(f)-r(f)+\\mathrm{mt}(f)$. For arbitrary characteristic, Deligne prove that there is always the inequality $\\mu(f)\\geq 2\\delta(f)-r(f)+1$ by showing that $\\mu(f)-\\left( 2\\delta(f)-r(f)+1\\right)$ measures the wild vanishing cycles. By introducing new invariants $\\gamma,\\tilde{\\gamma}$, we prove in this note that $\\kappa(f)\\geq \\gamma(f)+\\mathrm{mt}(f)-1\\geq 2\\delta(f)-r(f)+\\mathrm{mt}(f)$ with equalities if and only if the characteristic $p$ does not divide the multiplicity of any branch of $f$. As an application we show that if $p$ is \"big\" for $f$ (in fact $p > \\kappa(f)$), then $f$ has no wild vanishing cycle. Moreover we obtain some Pl\\\"ucker formulas for projective plane curves in positive characteristic.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-12-16T14:15:47.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "plane curve singularities",
      "positive characteristic",
      "plücker formulas",
      "wild vanishing cycle",
      "arbitrary characteristic"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1412.5007N"
    }
  }
}