{
  "id": "math/0406384",
  "version": "v1",
  "published": "2004-06-19T01:15:24.000Z",
  "updated": "2004-06-19T01:15:24.000Z",
  "title": "The affine stratification number and the moduli space of curves",
  "authors": [
    "Mike Roth",
    "Ravi Vakil"
  ],
  "comment": "17 pages, to appear in Proceedings of \"Workshop on algebraic structures and moduli spaces\", July 14-20, 2003, Centre de Recherches Mathematiques, Universite de Montreal",
  "categories": [
    "math.AG"
  ],
  "abstract": "We define the affine stratification number asn X of a scheme X. For X equidimensional, it is the minimal number k such that there is a stratification of X by locally closed affine subschemes of codimension at most k. We show that the affine stratification number is well-behaved, and bounds many aspects of the topological complexity of the scheme, such as vanishing of cohomology groups of quasicoherent, constructible, and l-adic sheaves. We explain how to bound asn X in practice. We give a series of conjectures (the first by E. Looijenga) bounding the affine stratification number of various moduli spaces of pointed curves. For example, the philosophy of [GV, Theorem *] yields: the moduli space of genus g, n-pointed complex curves of compact type (resp. with \"rational tails\") should have the homotopy type of a finite complex of dimension at most 5g-6+2n (resp. 4g-5+2n). This investigation is based on work and questions of Looijenga. One relevant example turns out to be a proper integral variety with no embeddings in a smooth algebraic space. This one-paragraph construction appears to be simpler and more elementary than the earlier examples, due to Horrocks and Nori.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2004-06-19T01:15:24.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14A15",
      "14H10"
    ],
    "keywords": [
      "moduli space",
      "affine stratification number asn",
      "one-paragraph construction appears",
      "smooth algebraic space",
      "proper integral variety"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 17,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2004math......6384R"
    }
  }
}