{
  "id": "1208.2898",
  "version": "v1",
  "published": "2012-08-14T15:31:28.000Z",
  "updated": "2012-08-14T15:31:28.000Z",
  "title": "A converse to a theorem of Oka and Sakamoto for complex line arrangements",
  "authors": [
    "Kristopher Williams"
  ],
  "comment": "15 pages, 3 figures",
  "categories": [
    "math.AG",
    "math.AT"
  ],
  "abstract": "Let C_1 and C_2 be algebraic plane curves in the complex plane such that the curves intersect in d_1\\cdot d_2 points where d_1,d_2 are the degrees of the curves respectively. Oka and Sakamoto proved that the fundamental group of the complement of C_1 \\cup C_2 is isomorphic to the direct of product of the fundamental group of the complement of C_1 and the fundamental group of the complement of C_2. In this paper we prove the converse of Oka and Sakamoto's result for line arrangements. Let A_1 and A_2 be non-empty arrangements of lines in complex plane such that the fundamental group of the complement of A_1 \\cup A_2 is isomorphic to the direct product of the complements of the arrangements A_1 and A_2. Then, the intersection of A_1 and A_2 consists of |A_1| \\cdot |A_2| points of multiplicity two.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2012-08-14T15:31:28.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F35",
      "57M05",
      "52C35"
    ],
    "keywords": [
      "complex line arrangements",
      "fundamental group",
      "complement",
      "complex plane",
      "algebraic plane curves"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 15,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2012arXiv1208.2898W"
    }
  }
}