{
  "id": "2207.14523",
  "version": "v1",
  "published": "2022-07-29T07:45:08.000Z",
  "updated": "2022-07-29T07:45:08.000Z",
  "title": "A note on the plane curve singularities in positive characteristic",
  "authors": [
    "Evelia R. García Barroso",
    "Arkadiusz Płoski"
  ],
  "comment": "15 pages, 3 figures",
  "categories": [
    "math.AG"
  ],
  "abstract": "Given an algebroid plane curve $f=0$ over an algebraically closed field of characteristic $p\\geq 0$ we consider the Milnor number $\\mu(f)$, the delta invariant $\\delta(f)$ and the number $r(f)$ of its irreducible components. Put $\\bar \\mu(f)=2\\delta(f)-r(f)+1$. If $p=0$ then $\\bar \\mu (f)=\\mu(f)$ (the Milnor formula). If $p>0$ then $\\mu(f)$ is not an invariant and $\\bar \\mu(f)$ plays the role of $\\mu(f)$. Let $\\mathcal N_f$ be the Newton polygon of $f$. We define the numbers $\\mu(\\mathcal N_{f})$ and $r(\\mathcal N_{f})$ which can be computed by explicit formulas. The aim of this note is to give a simple proof of the inequality $\\bar \\mu(f)-\\mu(\\mathcal N_{f})\\geq r(\\mathcal N_{f})- r(f)\\geq 0$ due to Boubakri, Greuel and Markwig. We also prove that $\\bar \\mu(f)=\\mu(\\mathcal N_{f})$ when $f$ is non-degenerate.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2022-07-29T07:45:08.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H20",
      "32S05"
    ],
    "keywords": [
      "plane curve singularities",
      "positive characteristic",
      "algebroid plane curve",
      "explicit formulas",
      "newton polygon"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}