{
  "id": "1402.2930",
  "version": "v2",
  "published": "2014-02-12T18:52:38.000Z",
  "updated": "2015-03-18T16:02:29.000Z",
  "title": "An Algorithm to Compute the Topological Euler Characteristic, Chern-Schwartz-MacPherson Class and Segre Class of Projective Varieties",
  "authors": [
    "Martin Helmer"
  ],
  "categories": [
    "math.AG"
  ],
  "abstract": "Let $V$ be a closed subscheme of a projective space $\\mathbb{P}^n$. We give an algorithm to compute the Chern-Schwartz-MacPherson class, Euler characteristic and Segre class of $ V$. The algorithm can be implemented using either symbolic or numerical methods. The algorithm is based on a new method for calculating the projective degrees of a rational map defined by a homogeneous ideal. Using this result and known formulas for the Chern-Schwartz-MacPherson class of a projective hypersurface and the Segre class of a projective variety in terms of the projective degrees of certain rational maps we give algorithms to compute the Chern-Schwartz-MacPherson class and Segre class of a projective variety. Since the Euler characteristic of $V$ is the degree of the zero dimensional component of the Chern-Schwartz-MacPherson class of $V$ our algorithm also computes the Euler characteristic $\\chi(V)$. Relationships between the algorithm developed here and other existing algorithms are discussed. The algorithm is tested on several examples and performs favourably compared to current algorithms for computing Chern-Schwartz-MacPherson classes, Segre classes and Euler characteristics.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2014-02-12T18:52:38.000Z",
      "comment": null,
      "journal": null,
      "doi": null
    },
    {
      "version": "v2",
      "updated": "2015-03-18T16:02:29.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14Qxx",
      "68W30",
      "14C17",
      "13P15",
      "65H10",
      "F.2.2",
      "I.1.2",
      "I.1.1"
    ],
    "keywords": [
      "projective variety",
      "topological euler characteristic",
      "rational map",
      "projective degrees",
      "zero dimensional component"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2014arXiv1402.2930H"
    }
  }
}