{
  "id": "math/0609561",
  "version": "v1",
  "published": "2006-09-20T14:05:17.000Z",
  "updated": "2006-09-20T14:05:17.000Z",
  "title": "Geometric collections and Castelnuovo-Mumford Regularity",
  "authors": [
    "L. Costa",
    "R. M. Miró-Roig"
  ],
  "comment": "To appear in Math. Proc. Cambridge",
  "categories": [
    "math.AG"
  ],
  "abstract": "The paper begins by overviewing the basic facts on geometric exceptional collections. Then, we derive, for any coherent sheaf $\\cF$ on a smooth projective variety with a geometric collection, two spectral sequences: the first one abuts to $\\cF$ and the second one to its cohomology. The main goal of the paper is to generalize Castelnuovo-Mumford regularity for coherent sheaves on projective spaces to coherent sheaves on smooth projective varieties $X$ with a geometric collection $\\sigma $. We define the notion of regularity of a coherent sheaf $\\cF$ on $X$ with respect to $\\sigma$. We show that the basic formal properties of the Castelnuovo-Mumford regularity of coherent sheaves over projective spaces continue to hold in this new setting and we show that in case of coherent sheaves on $\\PP^n$ and for a suitable geometric collection of coherent sheaves on $\\PP^n$ both notions of regularity coincide. Finally, we carefully study the regularity of coherent sheaves on a smooth quadric hypersurface $Q_n \\subset \\PP^{n+1}$ ($n$ odd) with respect to a suitable geometric collection and we compare it with the Castelnuovo-Mumford regularity of their extension by zero in $\\PP^{n+1}$.",
  "revisions": [
    {
      "version": "v1",
      "updated": "2006-09-20T14:05:17.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14F05",
      "18E30",
      "18F20"
    ],
    "keywords": [
      "coherent sheaves",
      "smooth projective variety",
      "suitable geometric collection",
      "coherent sheaf",
      "projective spaces"
    ],
    "note": {
      "typesetting": "TeX",
      "pages": 0,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "2006math......9561C"
    }
  }
}