{
  "id": "math/0104193",
  "version": "v1",
  "published": "2001-04-19T16:24:52.000Z",
  "updated": "2001-04-19T16:24:52.000Z",
  "title": "A Compactification of the Space of Plane Curves",
  "authors": [
    "Paul Hacking"
  ],
  "comment": "LaTeX, 74 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "We define a geometrically meaningful compactification of the moduli space of smooth plane curves, which can be calculated explicitly. The basic idea is to regard a plane curve D in P^2 as a pair (P^2,D) of a surface together with a divisor, and allow both the surface and the curve to degenerate. For plane curves of degree d at least 4, we obtain a compactification M_d which is a moduli space of stable pairs (X,D) using the log minimal model program. A stable pair (X,D) consists of a surface X such that -K_X is ample and a divisor D in a given linear system on X with specified singularities. Note that X may be non-normal, and K_X is Q-Cartier but not Cartier in general. We give a rough classification of stable pairs of arbitrary degree, a complete classification in degrees 4 and 5, and a partial classification in degree 6. The compactification is particularly simple if d is not a multiple of 3 - in particular the surface X has at most 2 components. We give a characterisation of these surfaces in terms of the singularities and the Picard numbers of the components. Moreover, we show that M_d is smooth in this case.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2001-04-19T16:24:52.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14H10",
      "14H50"
    ],
    "keywords": [
      "compactification",
      "stable pair",
      "moduli space",
      "log minimal model program",
      "smooth plane curves"
    ],
    "note": {
      "typesetting": "LaTeX",
      "pages": 74,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2001math......4193H"
    }
  }
}