{
  "id": "1004.1803",
  "version": "v2",
  "published": "2010-04-11T12:31:29.000Z",
  "updated": "2011-03-17T17:11:21.000Z",
  "title": "Monoidal transforms and invariants of singularities in positive characteristic",
  "authors": [
    "Angélica Benito",
    "Orlando E. Villamayor"
  ],
  "comment": "33 pages. This is a completely re-written version of the previous one. Many arguments are simplified and some parts will appear in a different manuscript",
  "categories": [
    "math.AG"
  ],
  "abstract": "The problem of resolution of singularities in positive characteristic can be reformulated as follows: Fix a hypersurface $X$, embedded in a smooth scheme, with points of multiplicity at most $n$. Let an $n$-sequence of transformations of $X$ be a finite composition of monoidal transformations with centers included in the $n$-fold points of $X$, and of its successive strict transforms. The open problem (in positive characteristic) is to prove that there is an $n$-sequence such that the final strict transform of $X$ has no points of multiplicity $n$ (no $n$-fold points). In characteristic zero, such an $n$-sequence is defined in two steps: the first consisting in the transformation of $X$ to a hypersurface with $n$-fold points in the so called monomial case. The second step consists in the elimination of these $n$-fold points (in the monomial case), which is achieved by a simple combinatorial procedure for choices of centers. The invariants treated in this work allow us to define a notion of strong monomial case which parallels that of monomial case in characteristic zero: If a hypersurface is within the strong monomial case we prove that a resolution can be achieved in a combinatorial manner.",
  "revisions": [
    {
      "version": "v2",
      "updated": "2011-03-17T17:11:21.000Z"
    }
  ],
  "analyses": {
    "keywords": [
      "positive characteristic",
      "fold points",
      "monoidal transforms",
      "strong monomial case",
      "invariants"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 33,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2010arXiv1004.1803B"
    }
  }
}