{
  "id": "1506.07996",
  "version": "v1",
  "published": "2015-06-26T08:49:54.000Z",
  "updated": "2015-06-26T08:49:54.000Z",
  "title": "Topologically equisingular deformations of homogeneous hypersurfaces with line singularities are equimultiple",
  "authors": [
    "Christophe Eyral"
  ],
  "categories": [
    "math.AG",
    "math.CV"
  ],
  "abstract": "We prove that if $\\{f_t\\}$ is a family of line singularities with constant L\\^e numbers and such that $f_0$ is a homogeneous polynomial, then $\\{f_t\\}$ is equimultiple. This extends to line singularities a well known theorem of A. M. Gabri\\`elov and A. G. Ku\\v{s}nirenko concerning isolated singularities. As an application, we show that if $\\{f_t\\}$ is a topologically $\\mathscr{V}$-equisingular family of line singularities, with $f_0$ homogeneous, then $\\{f_t\\}$ is equimultiple. This provides a new partial positive answer to the famous Zariski multiplicity conjecture for a special class of non-isolated hypersurface singularities.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2015-06-26T08:49:54.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "32S15",
      "32S25",
      "32S05"
    ],
    "keywords": [
      "line singularities",
      "topologically equisingular deformations",
      "homogeneous hypersurfaces",
      "equimultiple",
      "famous zariski multiplicity conjecture"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2015arXiv150607996E"
    }
  }
}