{
  "id": "math/9804114",
  "version": "v1",
  "published": "1998-04-23T11:41:34.000Z",
  "updated": "1998-04-23T11:41:34.000Z",
  "title": "Generic Projection Methods in Castelnuovo Regularity of Projective Varieties",
  "authors": [
    "Sijong Kwak"
  ],
  "comment": "AMSTeX; 18 pages",
  "categories": [
    "math.AG"
  ],
  "abstract": "Castelnuovo-Mumford regularity is an important invariant of projective algebraic varieties. A well known conjecture due to Eisenbud and Goto gives a bound for regularity in terms of the codimension and degree,i.e., Castelnuovo-Mumford regularity of a given variety X is less than or equal to deg(X)-codimension(X)+1. Generic projection methods proved to be effective for the study of regularity of smooth projevtive varieties of dimension at most four(cf.[BM},[K2],[L],[Pi] and [R1]) because there are nice vanishing theorems for cohomology of vector bundles (e.g. the Kodaira-Kawamata-Viehweg vanishing theorem) and detailed information about the fibers ofgeneric projections from X to a hypersurface of the same dimension. Here we show by using methods similar to those used in [K2] that $\\reg{X}\\le(deg(X)-codimension(X)+1)+10$ for any smooth fivefold and $\\reg{X}\\le(deg(X)-codimension(X)+1)+20$ for any smooth sixfold. Furthermore, using similar methods we give a bound for the regularity of an arbitrary (not necessarily locally Cohen-Macaulay) projective surface X in P^N. To wit, we show that $\\reg{X}\\le(d-e+1)d-(2e+1)$, where d=deg(X) and e=codimension(X). This is the first bound for surfaces which does not depend on smoothness.",
  "revisions": [
    {
      "version": "v1",
      "updated": "1998-04-23T11:41:34.000Z"
    }
  ],
  "analyses": {
    "subjects": [
      "14M07",
      "14N05"
    ],
    "keywords": [
      "generic projection methods",
      "castelnuovo regularity",
      "projective varieties",
      "castelnuovo-mumford regularity",
      "fibers ofgeneric projections"
    ],
    "note": {
      "typesetting": "AMS-TeX",
      "pages": 18,
      "language": "en",
      "license": "arXiv",
      "status": "editable",
      "adsabs": "1998math......4114K"
    }
  }
}