{
  "id": "1807.01153",
  "version": "v1",
  "published": "2018-07-03T13:21:19.000Z",
  "updated": "2018-07-03T13:21:19.000Z",
  "title": "On a resolution of singularities with two strata",
  "authors": [
    "Vincenzo Di Gennaro",
    "Davide Franco"
  ],
  "comment": "19 pages, no figures",
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $X$ be a complex, irreducible, quasi-projective variety, and $\\pi:\\widetilde X\\to X$ a resolution of singularities of $X$. Assume that the singular locus ${\\text{Sing}}(X)$ of $X$ is smooth, that the induced map $\\pi^{-1}({\\text{Sing}}(X))\\to {\\text{Sing}}(X)$ is a smooth fibration admitting a cohomology extension of the fiber, and that $\\pi^{-1}({\\text{Sing}}(X))$ has a negative normal bundle in $\\widetilde X$. We present a very short and explicit proof of the Decomposition Theorem for $\\pi$, providing a way to compute the intersection cohomology of $X$ by means of the cohomology of $\\widetilde X$ and of $\\pi^{-1}({\\text{Sing}}(X))$. Our result applies to special Schubert varieties with two strata, even if $\\pi$ is non-small. And to certain hypersurfaces of $\\mathbb P^5$ with one-dimensional singular locus.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2018-07-03T13:21:19.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14B05",
      "14E15",
      "14F05",
      "14F43",
      "14F45",
      "14M15",
      "32S20",
      "32S60",
      "58K15"
    ],
    "keywords": [
      "resolution",
      "singularities",
      "one-dimensional singular locus",
      "special schubert varieties",
      "cohomology extension"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 19,
      "language": "en",
      "license": "arXiv",
      "status": "editable"
    }
  }
}